##### Class 12^{th} Mathematics Part I CBSE Solution

**Exercise 3.1**- In a matrix A = a^n = [cccc 2&5&19&-7 35&-2& 5/2 &12 root 3 &1&-5&17] , (i) The…
- If a matrix has 24 elements, what are the possible orders it can have? What, if…
- If a matrix has 18 elements, what are the possible orders it can have? What, if…
- a_ij = (i+j)^2/2 Construct a 2 × 2 matrix, A = [aij], whose elements are given…
- a_ij = i/j Construct a 2 × 2 matrix, A = [aij], whose elements are given by:…
- a_ij = (i+2j)^2/2 Construct a 2 × 2 matrix, A = [aij], whose elements are given…
- a_ij = 1/2 |-3i+j| Construct a 3 × 4 matrix, whose elements are given by:…
- a_ij = 2i-j Construct a 3 × 4 matrix, whose elements are given by:…
- [ll 4&3 x&5] = [ll y 1&5] Find the values of x, y and z from the following…
- [cc x+y&2 5+z] = [ll 6&2 5&8] Find the values of x, y and z from the following…
- [c x+y+z x+z y+z] = [9 5 7] Find the values of x, y and z from the following…
- Find the value of a, b, c and d from the equation: [cc a-b&2a+c 2a-b&3c+d] = [cc…
- A = [aij]m × n is a square matrix, ifA. m n B. m n C. m = n D. None of these…
- Which of the given values of x and y make the following pair of matrices equal…
- The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is:A.…

**Exercise 3.2**- Le t a = [ll 2&4 3&2] , b = [cc 1&3 -2&5] , c = [cc -2&5 3&4] Find each of the…
- Let a = [ll 2&4 3&2] , b = [rr 1&3 -2&5] , c = [cc -2&5 3&4] Find each of the…
- Let a = [ll 2&4 3&2] , b = [rr 1&3 -2&5] , c = [cc -2&5 3&4] Find each of the…
- Let a = [ll 2&4 3&2] , b = [rr 1&3 -2&5] , c = [cc -2&5 3&4] Find each of the…
- Let a = [ll 2&4 3&2] , b = [rr 1&3 -2&5] , c = [cc -2&5 3&4] Find each of the…
- [cc a -b] + [ll a b] Compute the following:
- [cc a^2 + b^2 & b^2 + c^2 a^2 + c^2 & a^2 + b^2] + [cc 2ab&2bc -2ac&-2ab]…
- [ccc -1&4&-6 8&5&16 2&8&5] + [ccc 12&7&6 8&0&5 3&2&4] Compute the following:…
- [cc cos^2x^2x sin^2x^2x] + [cc sin^2x^2x cos^2x^2x] Compute the following:…
- Compute the indicated products.
- [1 2 3] [lll 2&3&4] Compute the indicated products.
- [cc 1&-2 2&3] [ccc 1&2&3 2&3&1] Compute the indicated products.
- [ccc 2&3&4 3&4&5 4&5&6] [ccc 1&-3&5 0&2&4 3&0&5] Compute the indicated…
- Compute the indicated products.
- [ccc 3&-1&3 -1&0&2] [cc 2&-3 1&0 3&1] Compute the indicated products.…
- If a = [ccc 1&2&-3 5&0&2 1&-1&1] , b = [ccc 3&-1&2 4&2&5 2&0&3] c = [ccc 4&1&2…
- If a = [ccc 2/3 &1& 5/3 1/3 & 2/3 & 4/3 7/3 &2& 2/3] b = [ccc 2/5 & 3/5 &1 1/5 &…
- Simplify costheta [cc costheta heta -sintegrate heta]+sintegrate heta [cc…
- x+y = [ll 7&0 2&5] x-y = [ll 3&0 0&3] Find X and Y, if
- 2x+3y = [ll 2&3 4&0] 3x+2y = [cc 2&-2 -1&5] Find X and Y, if
- Find X, if y = [ll 3&2 1&4] 2x+y = [cc 1&0 -3&2]
- Find x and y, if 2 [ll 1&3 0] + [ll y&0 1&2] = [ll 5&6 1&8]
- Solve the equation for x, y, z and t, if 2 [ll x y]+3 [cc 1&-1 0&2] = 3 [cc 3&5…
- If x [2 3]+y [c -1 1] = [c 10 5] , find the values of x and y.
- Given 3 [cc x z] = [cc x&6 -1&2w] + [cc 4+y z+w&3] , find the values of x, y, z…
- If f (x) = [ll cosx&-sinx&0 sinx&0 0&0&1 show that F(x) F(y) = F(x + y).…
- [cc 5&-1 6&7] [cc 2&1 3&4] not equal [ll 2&1 3&4] [cc 5&-1 6&7] Show that…
- [ccc 1&2&3 0&1&0 1&1&0] [ccc -1&1&0 0&-1&1 2&3&4] = [ccc -1&1&0 0&-1&1 2&3&4]…
- Find A^2 - 5A + 6I, if a = [lll 2&0&1 2&1&3 1&-1&0]
- If a = [lll 1&0&2 0&2&1 2&0&3] , prove that A3 - 6A^2 + 7A + 2I = 0…
- If a = [ll 3&-2 4&-2] i = [ll 1&0 0&1] , find k so that A^2 = kA - 2I…
- If a = [ccc 0& - tan alpha /2 tan alpha /2 &0] and I is the identity matrix of…
- Rs 1800 A trust fund has Rs. 30,000 that must be invested in two different…
- Rs. 2000 A trust fund has Rs. 30,000 that must be invested in two different…
- The bookshop of a particular school has 10 dozen chemistry books, 8 dozen…
- The restriction on n, k and p so that PY + WY will be defined are:A. k = 3, p =…
- If n = p, then the order of the matrix 7X - 5Z is:A. p × 2 B. 2 × n C. n × 3 D.…

**Exercise 3.3**- [c 5 1/2 -1] Find the transpose of each of the following matrices:…
- [cc 1&-1 2&3] Find the transpose of each of the following matrices:…
- [ccc -1&5&6 root 3 &5&6 2&3&-1] Find the transpose of each of the following…
- If a = [ccc -1&2&3 5&7&9 -2&1&1] and, then verify that (A + B)’ = A’ + B’,…
- If a = [ccc -1&2&3 5&7&9 -2&1&1] and, then verify that (A - B)’ = A’- B’…
- If a^there there eξ sts = [cc 3&4 -1&2 0&1] and, then verify that (A + B). = A’…
- If a^there there eξ sts = [cc 3&4 -1&2 0&1] and, then verify that (A - B)’ =…
- If a = [rr -2&3 1&2] and b = [rr -1&0 1&2] , then find (A + 2B)’
- (i) a = [c 1 -4 3] b = [lll -1&2&1] For the matrices A and B, verify that (AB)’…
- b = [lll 1&5&7]a = [0 1 2] For the matrices A and B, verify that (AB)’ = B’A’,…
- If a = [ll cosalpha -sinalpha] , then verify that A’ A = I
- If a = [cc sinalpha -cosalpha] , then verify that A’ A = I
- Show that the matrix a = [ccc 1&-1&5 -1&2&1 5&1&3] is a symmetric matrix.…
- Show that the matrix a = [ccc 0&1&-1 -1&0&1 1&-1&0] is a skew symmetric matrix.…
- For the matrix a = [ll 1&5 6&7] , verify that (A + A’) is a symmetric matrix…
- For the matrix a = [ll 1&5 6&7] , verify that (A - A’) is a skew symmetric…
- Find and 1/2 (a-a^there there eξ sts) , when a = [ccc 0 -a&0 -b&-c&0]…
- [cc 3&5 1&-1] Express the following matrices as the sum of a symmetric and a…
- [ccc 6&-2&2 -2&3&-1 2&-1&3] Express the following matrices as the sum of a…
- [ccc 3&3&-1 -2&-2&1 -4&-5&2] Express the following matrices as the sum of a…
- [cc 1&5 -1&2] [cc 3&5 1&-1] Express the following matrices as the sum of a…
- If A, B are symmetric matrices of same order, then AB - BA is aA. Skew…
- If , and A + A’ = I, if the value of a isA. 3 pi /2 B. pi /3 C. pi D. 3 pi /2…

**Exercise 3.4**- Using elementary transformations, find the inverse of each of the matrices.…
- [ll 2&1 1&1] Using elementary transformations, find the inverse of each of the…
- [ll 1&3 2&7] Using elementary transformations, find the inverse of each of the…
- [ll 2&3 5&7] Using elementary transformations, find the inverse of each of the…
- [ll 2&1 7&4] Using elementary transformations, find the inverse of each of the…
- [ll 2&5 1&3] Using elementary transformations, find the inverse of each of the…
- [ll 3&1 5&2] Using elementary transformations, find the inverse of each of the…
- [ll 4&5 3&4] Using elementary transformations, find the inverse of each of the…
- [cc 3&10 2&7] Using elementary transformations, find the inverse of each of the…
- [cc 3&-1 -4&2] Using elementary transformations, find the inverse of each of…
- [ll 2&-6 1&-2] Using elementary transformations, find the inverse of each of…
- [cc 6&-3 -2&1] Using elementary transformations, find the inverse of each of…
- [cc 2&-1 -3&2] Using elementary transformations, find the inverse of each of…
- [ll 2&1 4&2] Using elementary transformations, find the inverse of each of the…
- [ccc 2&-3&3 2&2&3 3&-2&2] Using elementary transformations, find the inverse of…
- [ccc 1&3&-2 -3&0&-5 2&5&0] Using elementary transformations, find the inverse…
- [ccc 2&0&-1 5&1&0 0&1&3] Using elementary transformations, find the inverse of…
- Matrices A and B will be inverse of each other only ifA. AB = BA B. AB = BA = 0…

**Miscellaneous Exercise**- Let, show that (aI + bA)n = anI + nan-1bA, where I is the identity matrix of…
- If a = [lll 1&1&1 1&1&1 1&1&1] , prove that .
- If a = [ll 3&-4 1&-1] then prove that a^n = [cc 1+2n&-4n n&1-2n] where n is any…
- If A and B are symmetric matrices, prove that AB - BA is a skew symmetric…
- Show that the matrix B’AB is symmetric or skew symmetric according as A is…
- Find the values of x, y, z if the matrix satisfy a = [ccc 0&2y x&-z x&-y] the…
- For what values of x: [lll 1&2&1] [lll 1&2&0 2&0&1 1&0&2] [0 2 x] = 0?…
- If a = [cc 3&1 -1&2] show that A^2 - 5A + 7I =0.
- Find x, if [rrr x&-5&-1] [lll 1&0&2 0&2&1 2&0&3] [x 4 1] = 0
- A manufacturer produces three products x, y, z which he sells in two markets.…
- Find the matrix X so that x [ccc 1&2&3 4&5&6] = [ccc -7&-8&-9 2&4&6]…
- If A and B are square matrices of the same order such that AB = BA, then prove…
- If a = [ll alpha & beta gamma & - alpha] is such that A � = I, thenA. 1 +…
- If the matrix A is both symmetric and skew symmetric, thenA. A is a diagonal…
- If A is square matrix such that A^2 = A, then (I + A)^3 7 A is equal toA. A B.…

**Exercise 3.1**

- In a matrix A = a^n = [cccc 2&5&19&-7 35&-2& 5/2 &12 root 3 &1&-5&17] , (i) The…
- If a matrix has 24 elements, what are the possible orders it can have? What, if…
- If a matrix has 18 elements, what are the possible orders it can have? What, if…
- a_ij = (i+j)^2/2 Construct a 2 × 2 matrix, A = [aij], whose elements are given…
- a_ij = i/j Construct a 2 × 2 matrix, A = [aij], whose elements are given by:…
- a_ij = (i+2j)^2/2 Construct a 2 × 2 matrix, A = [aij], whose elements are given…
- a_ij = 1/2 |-3i+j| Construct a 3 × 4 matrix, whose elements are given by:…
- a_ij = 2i-j Construct a 3 × 4 matrix, whose elements are given by:…
- [ll 4&3 x&5] = [ll y 1&5] Find the values of x, y and z from the following…
- [cc x+y&2 5+z] = [ll 6&2 5&8] Find the values of x, y and z from the following…
- [c x+y+z x+z y+z] = [9 5 7] Find the values of x, y and z from the following…
- Find the value of a, b, c and d from the equation: [cc a-b&2a+c 2a-b&3c+d] = [cc…
- A = [aij]m × n is a square matrix, ifA. m n B. m n C. m = n D. None of these…
- Which of the given values of x and y make the following pair of matrices equal…
- The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is:A.…

**Exercise 3.2**

- Le t a = [ll 2&4 3&2] , b = [cc 1&3 -2&5] , c = [cc -2&5 3&4] Find each of the…
- Let a = [ll 2&4 3&2] , b = [rr 1&3 -2&5] , c = [cc -2&5 3&4] Find each of the…
- Let a = [ll 2&4 3&2] , b = [rr 1&3 -2&5] , c = [cc -2&5 3&4] Find each of the…
- Let a = [ll 2&4 3&2] , b = [rr 1&3 -2&5] , c = [cc -2&5 3&4] Find each of the…
- Let a = [ll 2&4 3&2] , b = [rr 1&3 -2&5] , c = [cc -2&5 3&4] Find each of the…
- [cc a -b] + [ll a b] Compute the following:
- [cc a^2 + b^2 & b^2 + c^2 a^2 + c^2 & a^2 + b^2] + [cc 2ab&2bc -2ac&-2ab]…
- [ccc -1&4&-6 8&5&16 2&8&5] + [ccc 12&7&6 8&0&5 3&2&4] Compute the following:…
- [cc cos^2x^2x sin^2x^2x] + [cc sin^2x^2x cos^2x^2x] Compute the following:…
- Compute the indicated products.
- [1 2 3] [lll 2&3&4] Compute the indicated products.
- [cc 1&-2 2&3] [ccc 1&2&3 2&3&1] Compute the indicated products.
- [ccc 2&3&4 3&4&5 4&5&6] [ccc 1&-3&5 0&2&4 3&0&5] Compute the indicated…
- Compute the indicated products.
- [ccc 3&-1&3 -1&0&2] [cc 2&-3 1&0 3&1] Compute the indicated products.…
- If a = [ccc 1&2&-3 5&0&2 1&-1&1] , b = [ccc 3&-1&2 4&2&5 2&0&3] c = [ccc 4&1&2…
- If a = [ccc 2/3 &1& 5/3 1/3 & 2/3 & 4/3 7/3 &2& 2/3] b = [ccc 2/5 & 3/5 &1 1/5 &…
- Simplify costheta [cc costheta heta -sintegrate heta]+sintegrate heta [cc…
- x+y = [ll 7&0 2&5] x-y = [ll 3&0 0&3] Find X and Y, if
- 2x+3y = [ll 2&3 4&0] 3x+2y = [cc 2&-2 -1&5] Find X and Y, if
- Find X, if y = [ll 3&2 1&4] 2x+y = [cc 1&0 -3&2]
- Find x and y, if 2 [ll 1&3 0] + [ll y&0 1&2] = [ll 5&6 1&8]
- Solve the equation for x, y, z and t, if 2 [ll x y]+3 [cc 1&-1 0&2] = 3 [cc 3&5…
- If x [2 3]+y [c -1 1] = [c 10 5] , find the values of x and y.
- Given 3 [cc x z] = [cc x&6 -1&2w] + [cc 4+y z+w&3] , find the values of x, y, z…
- If f (x) = [ll cosx&-sinx&0 sinx&0 0&0&1 show that F(x) F(y) = F(x + y).…
- [cc 5&-1 6&7] [cc 2&1 3&4] not equal [ll 2&1 3&4] [cc 5&-1 6&7] Show that…
- [ccc 1&2&3 0&1&0 1&1&0] [ccc -1&1&0 0&-1&1 2&3&4] = [ccc -1&1&0 0&-1&1 2&3&4]…
- Find A^2 - 5A + 6I, if a = [lll 2&0&1 2&1&3 1&-1&0]
- If a = [lll 1&0&2 0&2&1 2&0&3] , prove that A3 - 6A^2 + 7A + 2I = 0…
- If a = [ll 3&-2 4&-2] i = [ll 1&0 0&1] , find k so that A^2 = kA - 2I…
- If a = [ccc 0& - tan alpha /2 tan alpha /2 &0] and I is the identity matrix of…
- Rs 1800 A trust fund has Rs. 30,000 that must be invested in two different…
- Rs. 2000 A trust fund has Rs. 30,000 that must be invested in two different…
- The bookshop of a particular school has 10 dozen chemistry books, 8 dozen…
- The restriction on n, k and p so that PY + WY will be defined are:A. k = 3, p =…
- If n = p, then the order of the matrix 7X - 5Z is:A. p × 2 B. 2 × n C. n × 3 D.…

**Exercise 3.3**

- [c 5 1/2 -1] Find the transpose of each of the following matrices:…
- [cc 1&-1 2&3] Find the transpose of each of the following matrices:…
- [ccc -1&5&6 root 3 &5&6 2&3&-1] Find the transpose of each of the following…
- If a = [ccc -1&2&3 5&7&9 -2&1&1] and, then verify that (A + B)’ = A’ + B’,…
- If a = [ccc -1&2&3 5&7&9 -2&1&1] and, then verify that (A - B)’ = A’- B’…
- If a^there there eξ sts = [cc 3&4 -1&2 0&1] and, then verify that (A + B). = A’…
- If a^there there eξ sts = [cc 3&4 -1&2 0&1] and, then verify that (A - B)’ =…
- If a = [rr -2&3 1&2] and b = [rr -1&0 1&2] , then find (A + 2B)’
- (i) a = [c 1 -4 3] b = [lll -1&2&1] For the matrices A and B, verify that (AB)’…
- b = [lll 1&5&7]a = [0 1 2] For the matrices A and B, verify that (AB)’ = B’A’,…
- If a = [ll cosalpha -sinalpha] , then verify that A’ A = I
- If a = [cc sinalpha -cosalpha] , then verify that A’ A = I
- Show that the matrix a = [ccc 1&-1&5 -1&2&1 5&1&3] is a symmetric matrix.…
- Show that the matrix a = [ccc 0&1&-1 -1&0&1 1&-1&0] is a skew symmetric matrix.…
- For the matrix a = [ll 1&5 6&7] , verify that (A + A’) is a symmetric matrix…
- For the matrix a = [ll 1&5 6&7] , verify that (A - A’) is a skew symmetric…
- Find and 1/2 (a-a^there there eξ sts) , when a = [ccc 0 -a&0 -b&-c&0]…
- [cc 3&5 1&-1] Express the following matrices as the sum of a symmetric and a…
- [ccc 6&-2&2 -2&3&-1 2&-1&3] Express the following matrices as the sum of a…
- [ccc 3&3&-1 -2&-2&1 -4&-5&2] Express the following matrices as the sum of a…
- [cc 1&5 -1&2] [cc 3&5 1&-1] Express the following matrices as the sum of a…
- If A, B are symmetric matrices of same order, then AB - BA is aA. Skew…
- If , and A + A’ = I, if the value of a isA. 3 pi /2 B. pi /3 C. pi D. 3 pi /2…

**Exercise 3.4**

- Using elementary transformations, find the inverse of each of the matrices.…
- [ll 2&1 1&1] Using elementary transformations, find the inverse of each of the…
- [ll 1&3 2&7] Using elementary transformations, find the inverse of each of the…
- [ll 2&3 5&7] Using elementary transformations, find the inverse of each of the…
- [ll 2&1 7&4] Using elementary transformations, find the inverse of each of the…
- [ll 2&5 1&3] Using elementary transformations, find the inverse of each of the…
- [ll 3&1 5&2] Using elementary transformations, find the inverse of each of the…
- [ll 4&5 3&4] Using elementary transformations, find the inverse of each of the…
- [cc 3&10 2&7] Using elementary transformations, find the inverse of each of the…
- [cc 3&-1 -4&2] Using elementary transformations, find the inverse of each of…
- [ll 2&-6 1&-2] Using elementary transformations, find the inverse of each of…
- [cc 6&-3 -2&1] Using elementary transformations, find the inverse of each of…
- [cc 2&-1 -3&2] Using elementary transformations, find the inverse of each of…
- [ll 2&1 4&2] Using elementary transformations, find the inverse of each of the…
- [ccc 2&-3&3 2&2&3 3&-2&2] Using elementary transformations, find the inverse of…
- [ccc 1&3&-2 -3&0&-5 2&5&0] Using elementary transformations, find the inverse…
- [ccc 2&0&-1 5&1&0 0&1&3] Using elementary transformations, find the inverse of…
- Matrices A and B will be inverse of each other only ifA. AB = BA B. AB = BA = 0…

**Miscellaneous Exercise**

- Let, show that (aI + bA)n = anI + nan-1bA, where I is the identity matrix of…
- If a = [lll 1&1&1 1&1&1 1&1&1] , prove that .
- If a = [ll 3&-4 1&-1] then prove that a^n = [cc 1+2n&-4n n&1-2n] where n is any…
- If A and B are symmetric matrices, prove that AB - BA is a skew symmetric…
- Show that the matrix B’AB is symmetric or skew symmetric according as A is…
- Find the values of x, y, z if the matrix satisfy a = [ccc 0&2y x&-z x&-y] the…
- For what values of x: [lll 1&2&1] [lll 1&2&0 2&0&1 1&0&2] [0 2 x] = 0?…
- If a = [cc 3&1 -1&2] show that A^2 - 5A + 7I =0.
- Find x, if [rrr x&-5&-1] [lll 1&0&2 0&2&1 2&0&3] [x 4 1] = 0
- A manufacturer produces three products x, y, z which he sells in two markets.…
- Find the matrix X so that x [ccc 1&2&3 4&5&6] = [ccc -7&-8&-9 2&4&6]…
- If A and B are square matrices of the same order such that AB = BA, then prove…
- If a = [ll alpha & beta gamma & - alpha] is such that A � = I, thenA. 1 +…
- If the matrix A is both symmetric and skew symmetric, thenA. A is a diagonal…
- If A is square matrix such that A^2 = A, then (I + A)^3 7 A is equal toA. A B.…

###### Exercise 3.1

**Question 1.**In a matrix A =

(i) The order of the matrix,

(ii) The number of elements,

(iii) Write the elements a_{13}, a_{21}, a_{33}, a_{24}, a_{23}.

**Answer:****(i)** In the given matrix, the number of rows is 3 and the number of columns is 4.

Order of a matrix =No of rows × No of columns

Therefore, the order of the matrix is 3 × 4.

**(ii)** Since, the order of the matrix is 3 × 4, there are 3 × 4 = 12 elements in it.

**(iii)** a_{13} = 19, a_{21} = 35, a_{33} = -5, a_{24} = 12, a_{23} = .

**Question 2.**If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?

**Answer:**It is known that if a matrix is of the order m × n, then it has mn elements.

Therefore, to find all the possible orders of a matrix having 24 elements, we had to find all the ordered pairs of natural numbers whose product is 24.

The ordered pairs are: (1, 24), (24, 1), (2, 12), (12, 2), (3, 8), (8, 3), (4, 6), and (6, 4).

Therefore, the possible orders of a matrix having 24 elements are;

1 × 24, 24 × 1, 2 × 12, 12 × 2, 3 × 8, 8 × 3, 4 × 6, 6 × 4

(1, 13) and (13, 1) are the ordered pairs of natural numbers whose product is 13.

Therefore, the possible orders of a matrix having 13 elements are 1 × 13 and 13 × 1.

**Question 3.**If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?

**Answer:**It is known that if a matrix s of the order m × n, then it has mn elements.

Therefore, to find all the possible orders of a matrix having 18 elements, we had to find all the ordered pairs of natural numbers whose product is 18.

The ordered pairs are: (1, 18), (18, 1), (2, 9), (9, 2), (3, 6) and (6, 3).

Therefore, the possible orders of a matrix having 24 elements are;

1 × 18, 18 × 1, 2 × 9, 9 × 2, 3 × 6, 6 × 3

(1, 5) and (5, 1) are the ordered pairs of natural numbers whose product is 5.

Therefore, the possible orders of a matrix having 5 elements are 1 × 5 and 5 × 1.

**Question 4.**Construct a 2 × 2 matrix, A = [a_{ij}], whose elements are given by:

**Answer:**In general, a 2 × 2 matrix is given by A =

a_{ij} =

Therefore,

a_{11} =

a_{12} =

a_{21} =

a_{22} =

Therefore, the required matrix is A =

**Question 5.**Construct a 2 × 2 matrix, A = [a_{ij}], whose elements are given by:

**Answer:**In general, a 2 × 2 matrix is given by A =

a_{ij} =

Therefore,

a_{11} =

a_{12} =

a_{21} =

a_{22} =

Therefore, the required matrix is A = .

**Question 6.**Construct a 2 × 2 matrix, A = [a_{ij}], whose elements are given by:

**Answer:**In general, a 2 × 2 matrix is given by A =

a_{ij} =

Therefore,

a_{11} =

a_{12} =

a_{21} =

a_{22} =

Therefore, the required matrix is A =

**Question 7.**Construct a 3 × 4 matrix, whose elements are given by:

**Answer:**In general 3 × 4 matrix is given by A =

Therefore,

a_{32} =

a_{13} =

a_{23} =

a_{33} =

a_{14} =

a_{24} =

a_{34} =

Therefore, required matrix is A =

**Question 8.**Construct a 3 × 4 matrix, whose elements are given by:

**Answer:**In general 3 × 4 matrix is given by A =

a_{ij} = 2i-j, i = 1,2,3 and j = 1,2,3,4

Therefore,

a_{11} = 2 × 1 - 1 = 2 - 1 = 1

a_{21} = 2 × 2 - 1 = 4 - 1 = 3

a_{31} = 2 × 3 - 1 = 6 - 1 = 5

a_{12} = 2 × 1 - 2 = 2 - 2 = 0

a_{22} = 2 × 2 - 2 = 4 - 2 = 2

a_{32} = 2 × 3 - 2 = 6 - 2 = 4

a_{13} = 2 × 1 - 3 = 2 - 3 = -1

a_{23} = 2 × 2 - 3 = 4 - 3 = 1

a_{33} = 2 × 3 - 3 = 6 - 3 = 3

a_{14} = 2 × 1 - 4 = 2 - 4 = -2

a_{24} = 2 × 2 - 4 = 4 - 4 = 0

a_{34} = 2 × 3 - 4 = 6 - 4 = 2

Therefore, required matrix is A =

**Question 9.**Find the values of x, y and z from the following equations:

**Answer:**Since, the given matrices are equal, their corresponding elements are also equal.

Comparing the corresponding element, we have:

x = 1, y = 4 and z = 3

**Question 10.**Find the values of x, y and z from the following equations:

**Answer:**Since, the given matrices are equal, their corresponding elements are also equal.

Comparing the corresponding element, we have:

x + y = 6, xy = 8, 5 + z = 5

Now, 5 + z = 5

⇒ z = 0

x+ y = 6

⇒ x = 6 - y

Also,

xy = 8

⇒ (6 - y)y = 8

⇒ 6y - y^{2}= 8

⇒ y^{2} - 6y + 8 = 0

⇒ y^{2} - 4y - 2y + 8 = 0

⇒ y(y - 4) - 2(y - 4) = 0

⇒ (y - 2)(y - 4) = 0

Hence, y = 2 or y = 4

when y = 2

x = 6 - 2 = 4

when x = 4

x = 6 - 4 = 2

**Question 11.**Find the values of x, y and z from the following equations:

**Answer:**Since, the given matrices are equal, their corresponding elements are also equal.

Comparing the corresponding element, we have:

x + y + z = 9...(1)

x + z = 5.....(2)

y + z = 7.......(3)

Putting the value of equation 2 in equation 1,

y + 5 = 9

⇒ y = 4

Then, putting the value of y in equation 3, we get,

4 + z = 7

⇒ z = 3

Therefore, x + z = 5

⇒ x = 2

Therefore, x =2, y = 4 and z = 3.

**Question 12.**Find the value of a, b, c and d from the equation:

**Answer:**

Since, the given matrices are equal, their corresponding elements are also equal.

Comparing the corresponding element, we have:

a –b = -1 …(1)

2a – b = 0 …(2)

2a + c = 5 …(3)

3c + d = 13 …(4)

From equation (2), we get:

b = 2a

Then, from eq. (1), we get,

a -2a = -1

⇒ a = 1

⇒ b = 2

Now, from eq. (3), we get:

2 × 1 + c = 5

⇒ c = 3

From (4), we get,

3 × 3 + d = 13

⇒ 9 + d = 13

⇒ d = 4

Therefore, a = 1, b = 2, c = 3 and d = 4.

**Question 13.**A = [a_{ij}]_{m × n} is a square matrix, if

A. m < n

B. m > n

C. m = n

D. None of these

**Answer:**We know that if a given matrix is said to be square matrix if the number of rows is equal to the number of columns.

Therefore, A = [a_{ij}]_{m × n} is a square matrix, if m = n.

**Question 14.**Which of the given values of x and y make the following pair of matrices equal

A.

B. Not possible to find

C. y = 7,

D.

**Answer:**Now,

Since, the given matrices are equal, their corresponding elements are also equal.

Comparing the corresponding element, we have:

3x + 7 = 0

5 = y – 2

⇒ y = 7

y + 1 = 8

⇒ y = 7

And 2 – 3x = 4

Thus, on comparing the corresponding elements of the two matrices, we get different values of x, which is not possible.

Therefore, it is not possible to find the values of x and y for which the given matrices are equal.

**Question 15.**The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is:

A. 27

B. 18

C. 81

D. 512

**Answer:**The given matrix of the order 3 × 3 has 9 elements with each entry 0 or 1.

Now, each of the 9 elements can be filled in two possible ways.

Hence, the required number of possible matrices is 2^{9} = 512.

**Question 1.**

In a matrix A =

(i) The order of the matrix,

(ii) The number of elements,

(iii) Write the elements a_{13}, a_{21}, a_{33}, a_{24}, a_{23}.

**Answer:**

**(i)** In the given matrix, the number of rows is 3 and the number of columns is 4.

Therefore, the order of the matrix is 3 × 4.

**(ii)** Since, the order of the matrix is 3 × 4, there are 3 × 4 = 12 elements in it.

**(iii)** a_{13} = 19, a_{21} = 35, a_{33} = -5, a_{24} = 12, a_{23} = .

**Question 2.**

If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?

**Answer:**

It is known that if a matrix is of the order m × n, then it has mn elements.

Therefore, to find all the possible orders of a matrix having 24 elements, we had to find all the ordered pairs of natural numbers whose product is 24.

The ordered pairs are: (1, 24), (24, 1), (2, 12), (12, 2), (3, 8), (8, 3), (4, 6), and (6, 4).

Therefore, the possible orders of a matrix having 24 elements are;

1 × 24, 24 × 1, 2 × 12, 12 × 2, 3 × 8, 8 × 3, 4 × 6, 6 × 4

(1, 13) and (13, 1) are the ordered pairs of natural numbers whose product is 13.

Therefore, the possible orders of a matrix having 13 elements are 1 × 13 and 13 × 1.

**Question 3.**

If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?

**Answer:**

It is known that if a matrix s of the order m × n, then it has mn elements.

Therefore, to find all the possible orders of a matrix having 18 elements, we had to find all the ordered pairs of natural numbers whose product is 18.

The ordered pairs are: (1, 18), (18, 1), (2, 9), (9, 2), (3, 6) and (6, 3).

Therefore, the possible orders of a matrix having 24 elements are;

1 × 18, 18 × 1, 2 × 9, 9 × 2, 3 × 6, 6 × 3

(1, 5) and (5, 1) are the ordered pairs of natural numbers whose product is 5.

Therefore, the possible orders of a matrix having 5 elements are 1 × 5 and 5 × 1.

**Question 4.**

Construct a 2 × 2 matrix, A = [a_{ij}], whose elements are given by:

**Answer:**

In general, a 2 × 2 matrix is given by A =

a_{ij} =

Therefore,

a_{11} =

a_{12} =

a_{21} =

a_{22} =

Therefore, the required matrix is A =

**Question 5.**

Construct a 2 × 2 matrix, A = [a_{ij}], whose elements are given by:

**Answer:**

In general, a 2 × 2 matrix is given by A =

a_{ij} =

Therefore,

a_{11} =

a_{12} =

a_{21} =

a_{22} =

Therefore, the required matrix is A = .

**Question 6.**

Construct a 2 × 2 matrix, A = [a_{ij}], whose elements are given by:

**Answer:**

In general, a 2 × 2 matrix is given by A =

a_{ij} =

Therefore,

a_{11} =

a_{12} =

a_{21} =

a_{22} =

Therefore, the required matrix is A =

**Question 7.**

Construct a 3 × 4 matrix, whose elements are given by:

**Answer:**

In general 3 × 4 matrix is given by A =

Therefore,

a_{32} =

a_{13} =

a_{23} =

a_{33} =

a_{14} =

a_{24} =

a_{34} =

Therefore, required matrix is A =

**Question 8.**

Construct a 3 × 4 matrix, whose elements are given by:

**Answer:**

In general 3 × 4 matrix is given by A =

a_{ij} = 2i-j, i = 1,2,3 and j = 1,2,3,4

Therefore,

a_{11} = 2 × 1 - 1 = 2 - 1 = 1

a_{21} = 2 × 2 - 1 = 4 - 1 = 3

a_{31} = 2 × 3 - 1 = 6 - 1 = 5

a_{12} = 2 × 1 - 2 = 2 - 2 = 0

a_{22} = 2 × 2 - 2 = 4 - 2 = 2

a_{32} = 2 × 3 - 2 = 6 - 2 = 4

a_{13} = 2 × 1 - 3 = 2 - 3 = -1

a_{23} = 2 × 2 - 3 = 4 - 3 = 1

a_{33} = 2 × 3 - 3 = 6 - 3 = 3

a_{14} = 2 × 1 - 4 = 2 - 4 = -2

a_{24} = 2 × 2 - 4 = 4 - 4 = 0

a_{34} = 2 × 3 - 4 = 6 - 4 = 2

Therefore, required matrix is A =

**Question 9.**

Find the values of x, y and z from the following equations:

**Answer:**

Since, the given matrices are equal, their corresponding elements are also equal.

Comparing the corresponding element, we have:

x = 1, y = 4 and z = 3

**Question 10.**

Find the values of x, y and z from the following equations:

**Answer:**

Since, the given matrices are equal, their corresponding elements are also equal.

Comparing the corresponding element, we have:

x + y = 6, xy = 8, 5 + z = 5

Now, 5 + z = 5

⇒ z = 0

x+ y = 6

⇒ x = 6 - y

Also,

xy = 8

⇒ (6 - y)y = 8

⇒ 6y - y

^{2}= 8

⇒ y

^{2}- 6y + 8 = 0

⇒ y

^{2}- 4y - 2y + 8 = 0

⇒ y(y - 4) - 2(y - 4) = 0

⇒ (y - 2)(y - 4) = 0

Hence, y = 2 or y = 4

when y = 2

x = 6 - 2 = 4

when x = 4

x = 6 - 4 = 2

**Question 11.**

Find the values of x, y and z from the following equations:

**Answer:**

Since, the given matrices are equal, their corresponding elements are also equal.

Comparing the corresponding element, we have:

x + y + z = 9...(1)

x + z = 5.....(2)

y + z = 7.......(3)

Putting the value of equation 2 in equation 1,y + 5 = 9

⇒ y = 4

Then, putting the value of y in equation 3, we get,

4 + z = 7

⇒ z = 3

Therefore, x + z = 5

⇒ x = 2

Therefore, x =2, y = 4 and z = 3.

**Question 12.**

Find the value of a, b, c and d from the equation:

**Answer:**

Since, the given matrices are equal, their corresponding elements are also equal.

Comparing the corresponding element, we have:

a –b = -1 …(1)

2a – b = 0 …(2)

2a + c = 5 …(3)

3c + d = 13 …(4)

From equation (2), we get:

b = 2a

Then, from eq. (1), we get,

a -2a = -1

⇒ a = 1

⇒ b = 2

Now, from eq. (3), we get:

2 × 1 + c = 5

⇒ c = 3

From (4), we get,

3 × 3 + d = 13

⇒ 9 + d = 13

⇒ d = 4

Therefore, a = 1, b = 2, c = 3 and d = 4.

**Question 13.**

A = [a_{ij}]_{m × n} is a square matrix, if

A. m < n

B. m > n

C. m = n

D. None of these

**Answer:**

We know that if a given matrix is said to be square matrix if the number of rows is equal to the number of columns.

Therefore, A = [a_{ij}]_{m × n} is a square matrix, if m = n.

**Question 14.**

Which of the given values of x and y make the following pair of matrices equal

A.

B. Not possible to find

C. y = 7,

D.

**Answer:**

Now,

Since, the given matrices are equal, their corresponding elements are also equal.

Comparing the corresponding element, we have:

3x + 7 = 0

5 = y – 2

⇒ y = 7

y + 1 = 8

⇒ y = 7

And 2 – 3x = 4

Thus, on comparing the corresponding elements of the two matrices, we get different values of x, which is not possible.

Therefore, it is not possible to find the values of x and y for which the given matrices are equal.

**Question 15.**

The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is:

A. 27

B. 18

C. 81

D. 512

**Answer:**

The given matrix of the order 3 × 3 has 9 elements with each entry 0 or 1.

Now, each of the 9 elements can be filled in two possible ways.

Hence, the required number of possible matrices is 2^{9} = 512.

###### Exercise 3.2

**Question 1.**Le t

Find each of the following:

i) A + B

ii) A - B

iii) 3A - C

iv) AB

v) BA

**Answer:****i)** A + B =

**ii)** A - B

**iii)** 3A - C

**iv)** AB

**v)** BA

**Question 2.**Let

Find each of the following:

A – B

**Answer:**A – B =

**Question 3.**Let

Find each of the following:

3A – C

**Answer:**3A – C =

**Question 4.**Let

Find each of the following:

AB

**Answer:**AB

**Question 5.**Let

Find each of the following:

BA

**Answer:**BA =

**Question 6.**Compute the following:

**Answer:**

**Question 7.**Compute the following:

**Answer:**

.

**Question 8.**Compute the following:

**Answer:**

**Question 9.**Compute the following:

**Answer:**

**Question 10.**Compute the indicated products.

**Answer:**

**Question 11.**Compute the indicated products.

**Answer:**

**Question 12.**Compute the indicated products.

**Answer:**

**Question 13.**Compute the indicated products.

**Answer:**

.

.

**Question 14.**Compute the indicated products.

**Answer:**

**Question 15.**Compute the indicated products.

**Answer:**

**Question 16.**If then compute (A + B) and (B – C). Also, verify that A + (B – C) = (A + B) – C.

**Answer:**Now, A+ B

B – C

A + (B –C)

.

+ B) – C =

.

Therefore, A + (B –C) = (A + B) – C

**Question 17.**If then compute 3A – 5B.

**Answer:**3A – 5B

**Question 18.**Simplify

**Answer:**

.

**Question 19.**Find X and Y, if

**Answer:**Now, X + Y = …(1)

Adding (1) and (2), we get,

2X =

.

Now, X + Y =

**Question 20.**Find X and Y, if

**Answer:**2X + 3Y …(1)

3X + 2Y …(2)

Now, multiply equation (1) by 2 and equation (2) by 3, we get,

4X + 6Y …(3)

9X + 6Y …(4)

Subtracting equation (4) from (3), we get,

(4X + 6Y) – (9X + 6Y)

Now, 2X + 3Y

**Question 21.**Find X, if

**Answer:**2X + Y

**Question 22.**Find x and y, if

**Answer:**

Now, on comparing elements of these two matrices, we get,

2 + y = 5

=> y = 3

And 2x + 2 = 8

=> x = 3

Therefore, x = 3 and y = 3.

**Question 23.**Solve the equation for x, y, z and t, if

**Answer:**

On comparing the elements of these two matrices, we get,

2x + 3 = 9

⇒ 2x = 6

⇒ x = 3

2y = 12

⇒ y = 6

2z -3 = 15

⇒ 2z = 18

⇒ z = 9

2t +6 = 18

⇒ 2t = 12

⇒ t = 6

Therefore, x = 3, y =6, z = 9 and t = 6.

**Question 24.**If , find the values of x and y.

**Answer:**

On comparing the corresponding elements of these two matrices, we get,

2x –y = 10 and 3x + y =5

Now, adding above two equations, we get

5x = 15

⇒ x = 3

Now, 3x + y = 5

⇒ y = 5 - 3x

⇒ y = 5 - 9 = -4

Therefore, x = 3 and y = -4.

**Question 25.**Given , find the values of x, y, z and w.

**Answer:**

On comparing the corresponding elements of these two matrices, we get,

3x = x + 4

⇒ 2x + 4

⇒ x =2

3y = 6 + x + y

⇒ 2y = 6 + x = 6 + 2 = 8

⇒ y = 4

3w = 2w + 3

⇒ w = 3

3z = -1 + z + w

⇒ 2z = -1 + w = -1 +3 = 2

⇒ z = 1

Therefore, x = 2, y = 4, z = 1 and w = 3.

**Question 26.**If show that F(x) F(y) = F(x + y).

**Answer:**

Therefore, F(x)F(y) = F(x+y)

**Question 27.**Show that

**Answer:**

**Question 28.**Show that

**Answer:**

**Question 29.**Find A^{2} – 5A + 6I, if

**Answer:**A^{2} = A.A

Now, A^{2} – 5A + 6I

**Question 30.**If, prove that A_{3} – 6A^{2} + 7A + 2I = 0

**Answer:**A^{2} = A.A

Now, A^{3} = A^{2}. A

Now, A^{3} – 6A^{2} + 7A + 2I

Therefore, A^{3} – 6A^{2} + 7A + 2I = 0

**Question 31.**If , find k so that A^{2} = kA – 2I

**Answer:**A^{2} = A.A

Now, A^{2} = kA – 2I

Comparing the corresponding elements, we get,

3k -2 = 1

⇒ 3k = 3

⇒ k = 1

Therefore, the value of k is 1.

**Question 32.**If and I is the identity matrix of order 2, show that

**Answer:**We know that I

Now, LHS = I + A

=

And on RHS = (I – A)

As we know,

cos 2θ = 1 - 2 sin^{2} θ

cos 2θ = 2 cos^{2} θ - 1

and sin2θ = 2 sinθ cosθ

As tanθ = sinθ/cosθ

=

= LHS

Hence Proved.

**Question 33.**A trust fund has Rs. 30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs. 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of:

Rs 1800

**Answer:**(a) Let Rs x be invested in the first bond.

Then, the sum of money invested in the second bond will be Rs. (30000 – x).

It is given that the first bond pays 5% interest per year, and the second bond pays 7% interest per year.

Thus, in order to obtain an annual total interest of Rs. 1800, we get:

**Question 1.**

Le t

Find each of the following:

i) A + B

ii) A - B

iii) 3A - C

iv) AB

v) BA

**Answer:**

**i)** A + B =

**ii)** A - B**iii)** 3A - C**iv)** AB**v)** BA

**Question 2.**

Let

Find each of the following:

A – B

**Answer:**

A – B =

**Question 3.**

Let

Find each of the following:

3A – C

**Answer:**

3A – C =

**Question 4.**

Let

Find each of the following:

AB

**Answer:**

AB

**Question 5.**

Let

Find each of the following:

BA

**Answer:**

BA =

**Question 6.**

Compute the following:

**Answer:**

**Question 7.**

Compute the following:

**Answer:**

.

**Question 8.**

Compute the following:

**Answer:**

**Question 9.**

Compute the following:

**Answer:**

**Question 10.**

Compute the indicated products.

**Answer:**

**Question 11.**

Compute the indicated products.

**Answer:**

**Question 12.**

Compute the indicated products.

**Answer:**

**Question 13.**

Compute the indicated products.

**Answer:**

.

.

**Question 14.**

Compute the indicated products.

**Answer:**

**Question 15.**

Compute the indicated products.

**Answer:**

**Question 16.**

If then compute (A + B) and (B – C). Also, verify that A + (B – C) = (A + B) – C.

**Answer:**

Now, A+ B

B – C

A + (B –C)

.

+ B) – C =

.

Therefore, A + (B –C) = (A + B) – C

**Question 17.**

If then compute 3A – 5B.

**Answer:**

3A – 5B

**Question 18.**

Simplify

**Answer:**

.

**Question 19.**

Find X and Y, if

**Answer:**

Now, X + Y = …(1)

Adding (1) and (2), we get,

2X =

.

Now, X + Y =

**Question 20.**

Find X and Y, if

**Answer:**

2X + 3Y …(1)

3X + 2Y …(2)

Now, multiply equation (1) by 2 and equation (2) by 3, we get,

4X + 6Y …(3)

9X + 6Y …(4)

Subtracting equation (4) from (3), we get,

(4X + 6Y) – (9X + 6Y)

Now, 2X + 3Y

**Question 21.**

Find X, if

**Answer:**

2X + Y

**Question 22.**

Find x and y, if

**Answer:**

Now, on comparing elements of these two matrices, we get,

2 + y = 5

=> y = 3

And 2x + 2 = 8

=> x = 3

Therefore, x = 3 and y = 3.

**Question 23.**

Solve the equation for x, y, z and t, if

**Answer:**

On comparing the elements of these two matrices, we get,

2x + 3 = 9

⇒ 2x = 6

⇒ x = 3

2y = 12

⇒ y = 6

2z -3 = 15

⇒ 2z = 18

⇒ z = 9

2t +6 = 18

⇒ 2t = 12

⇒ t = 6

Therefore, x = 3, y =6, z = 9 and t = 6.

**Question 24.**

If , find the values of x and y.

**Answer:**

On comparing the corresponding elements of these two matrices, we get,

2x –y = 10 and 3x + y =5

Now, adding above two equations, we get

5x = 15

⇒ x = 3

Now, 3x + y = 5

⇒ y = 5 - 3x

⇒ y = 5 - 9 = -4

Therefore, x = 3 and y = -4.

**Question 25.**

Given , find the values of x, y, z and w.

**Answer:**

On comparing the corresponding elements of these two matrices, we get,

3x = x + 4

⇒ 2x + 4

⇒ x =2

3y = 6 + x + y

⇒ 2y = 6 + x = 6 + 2 = 8

⇒ y = 4

3w = 2w + 3

⇒ w = 3

3z = -1 + z + w

⇒ 2z = -1 + w = -1 +3 = 2

⇒ z = 1

Therefore, x = 2, y = 4, z = 1 and w = 3.

**Question 26.**

If show that F(x) F(y) = F(x + y).

**Answer:**

Therefore, F(x)F(y) = F(x+y)

**Question 27.**

Show that

**Answer:**

**Question 28.**

Show that

**Answer:**

**Question 29.**

Find A^{2} – 5A + 6I, if

**Answer:**

A^{2} = A.A

Now, A^{2} – 5A + 6I

**Question 30.**

If, prove that A_{3} – 6A^{2} + 7A + 2I = 0

**Answer:**

A^{2} = A.A

Now, A^{3} = A^{2}. A

Now, A^{3} – 6A^{2} + 7A + 2I

Therefore, A^{3} – 6A^{2} + 7A + 2I = 0

**Question 31.**

If , find k so that A^{2} = kA – 2I

**Answer:**

A^{2} = A.A

Now, A^{2} = kA – 2I

Comparing the corresponding elements, we get,

3k -2 = 1

⇒ 3k = 3

⇒ k = 1

Therefore, the value of k is 1.

**Question 32.**

If and I is the identity matrix of order 2, show that

**Answer:**

We know that I

Now, LHS = I + A

=

And on RHS = (I – A)

As we know,

cos 2θ = 1 - 2 sin^{2} θ

cos 2θ = 2 cos^{2} θ - 1

and sin2θ = 2 sinθ cosθ

As tanθ = sinθ/cosθ

=

= LHS

Hence Proved.

**Question 33.**

A trust fund has Rs. 30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs. 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of:

Rs 1800

**Answer:**

(a) Let Rs x be invested in the first bond.

Then, the sum of money invested in the second bond will be Rs. (30000 – x).

It is given that the first bond pays 5% interest per year, and the second bond pays 7% interest per year.

Thus, in order to obtain an annual total interest of Rs. 1800, we get: