Matrices Class 12th Mathematics Part I CBSE Solution

Class 12th Mathematics Part I CBSE Solution
Exercise 3.1
  1. In a matrix A = a^n = [cccc 2&5&19&-7 35&-2& 5/2 &12 root 3 &1&-5&17] , (i) The…
  2. If a matrix has 24 elements, what are the possible orders it can have? What, if…
  3. If a matrix has 18 elements, what are the possible orders it can have? What, if…
  4. a_ij = (i+j)^2/2 Construct a 2 × 2 matrix, A = [aij], whose elements are given…
  5. a_ij = i/j Construct a 2 × 2 matrix, A = [aij], whose elements are given by:…
  6. a_ij = (i+2j)^2/2 Construct a 2 × 2 matrix, A = [aij], whose elements are given…
  7. a_ij = 1/2 |-3i+j| Construct a 3 × 4 matrix, whose elements are given by:…
  8. a_ij = 2i-j Construct a 3 × 4 matrix, whose elements are given by:…
  9. [ll 4&3 x&5] = [ll y 1&5] Find the values of x, y and z from the following…
  10. [cc x+y&2 5+z] = [ll 6&2 5&8] Find the values of x, y and z from the following…
  11. [c x+y+z x+z y+z] = [9 5 7] Find the values of x, y and z from the following…
  12. Find the value of a, b, c and d from the equation: [cc a-b&2a+c 2a-b&3c+d] = [cc…
  13. A = [aij]m × n is a square matrix, ifA. m n B. m n C. m = n D. None of these…
  14. Which of the given values of x and y make the following pair of matrices equal…
  15. The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is:A.…
Exercise 3.2
  1. Le t a = [ll 2&4 3&2] , b = [cc 1&3 -2&5] , c = [cc -2&5 3&4] Find each of the…
  2. Let a = [ll 2&4 3&2] , b = [rr 1&3 -2&5] , c = [cc -2&5 3&4] Find each of the…
  3. Let a = [ll 2&4 3&2] , b = [rr 1&3 -2&5] , c = [cc -2&5 3&4] Find each of the…
  4. Let a = [ll 2&4 3&2] , b = [rr 1&3 -2&5] , c = [cc -2&5 3&4] Find each of the…
  5. Let a = [ll 2&4 3&2] , b = [rr 1&3 -2&5] , c = [cc -2&5 3&4] Find each of the…
  6. [cc a -b] + [ll a b] Compute the following:
  7. [cc a^2 + b^2 & b^2 + c^2 a^2 + c^2 & a^2 + b^2] + [cc 2ab&2bc -2ac&-2ab]…
  8. [ccc -1&4&-6 8&5&16 2&8&5] + [ccc 12&7&6 8&0&5 3&2&4] Compute the following:…
  9. [cc cos^2x^2x sin^2x^2x] + [cc sin^2x^2x cos^2x^2x] Compute the following:…
  10. Compute the indicated products.
  11. [1 2 3] [lll 2&3&4] Compute the indicated products.
  12. [cc 1&-2 2&3] [ccc 1&2&3 2&3&1] Compute the indicated products.
  13. [ccc 2&3&4 3&4&5 4&5&6] [ccc 1&-3&5 0&2&4 3&0&5] Compute the indicated…
  14. Compute the indicated products.
  15. [ccc 3&-1&3 -1&0&2] [cc 2&-3 1&0 3&1] Compute the indicated products.…
  16. 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…
  17. 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 &…
  18. Simplify costheta [cc costheta heta -sintegrate heta]+sintegrate heta [cc…
  19. x+y = [ll 7&0 2&5] x-y = [ll 3&0 0&3] Find X and Y, if
  20. 2x+3y = [ll 2&3 4&0] 3x+2y = [cc 2&-2 -1&5] Find X and Y, if
  21. Find X, if y = [ll 3&2 1&4] 2x+y = [cc 1&0 -3&2]
  22. Find x and y, if 2 [ll 1&3 0] + [ll y&0 1&2] = [ll 5&6 1&8]
  23. Solve the equation for x, y, z and t, if 2 [ll x y]+3 [cc 1&-1 0&2] = 3 [cc 3&5…
  24. If x [2 3]+y [c -1 1] = [c 10 5] , find the values of x and y.
  25. Given 3 [cc x z] = [cc x&6 -1&2w] + [cc 4+y z+w&3] , find the values of x, y, z…
  26. If f (x) = [ll cosx&-sinx&0 sinx&0 0&0&1 show that F(x) F(y) = F(x + y).…
  27. [cc 5&-1 6&7] [cc 2&1 3&4] not equal [ll 2&1 3&4] [cc 5&-1 6&7] Show that…
  28. [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]…
  29. Find A^2 - 5A + 6I, if a = [lll 2&0&1 2&1&3 1&-1&0]
  30. If a = [lll 1&0&2 0&2&1 2&0&3] , prove that A3 - 6A^2 + 7A + 2I = 0…
  31. If a = [ll 3&-2 4&-2] i = [ll 1&0 0&1] , find k so that A^2 = kA - 2I…
  32. If a = [ccc 0& - tan alpha /2 tan alpha /2 &0] and I is the identity matrix of…
  33. Rs 1800 A trust fund has Rs. 30,000 that must be invested in two different…
  34. Rs. 2000 A trust fund has Rs. 30,000 that must be invested in two different…
  35. The bookshop of a particular school has 10 dozen chemistry books, 8 dozen…
  36. The restriction on n, k and p so that PY + WY will be defined are:A. k = 3, p =…
  37. 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
  1. [c 5 1/2 -1] Find the transpose of each of the following matrices:…
  2. [cc 1&-1 2&3] Find the transpose of each of the following matrices:…
  3. [ccc -1&5&6 root 3 &5&6 2&3&-1] Find the transpose of each of the following…
  4. If a = [ccc -1&2&3 5&7&9 -2&1&1] and, then verify that (A + B)’ = A’ + B’,…
  5. If a = [ccc -1&2&3 5&7&9 -2&1&1] and, then verify that (A - B)’ = A’- B’…
  6. If a^there there eξ sts = [cc 3&4 -1&2 0&1] and, then verify that (A + B). = A’…
  7. If a^there there eξ sts = [cc 3&4 -1&2 0&1] and, then verify that (A - B)’ =…
  8. If a = [rr -2&3 1&2] and b = [rr -1&0 1&2] , then find (A + 2B)’
  9. (i) a = [c 1 -4 3] b = [lll -1&2&1] For the matrices A and B, verify that (AB)’…
  10. b = [lll 1&5&7]a = [0 1 2] For the matrices A and B, verify that (AB)’ = B’A’,…
  11. If a = [ll cosalpha -sinalpha] , then verify that A’ A = I
  12. If a = [cc sinalpha -cosalpha] , then verify that A’ A = I
  13. Show that the matrix a = [ccc 1&-1&5 -1&2&1 5&1&3] is a symmetric matrix.…
  14. Show that the matrix a = [ccc 0&1&-1 -1&0&1 1&-1&0] is a skew symmetric matrix.…
  15. For the matrix a = [ll 1&5 6&7] , verify that (A + A’) is a symmetric matrix…
  16. For the matrix a = [ll 1&5 6&7] , verify that (A - A’) is a skew symmetric…
  17. Find and 1/2 (a-a^there there eξ sts) , when a = [ccc 0 -a&0 -b&-c&0]…
  18. [cc 3&5 1&-1] Express the following matrices as the sum of a symmetric and a…
  19. [ccc 6&-2&2 -2&3&-1 2&-1&3] Express the following matrices as the sum of a…
  20. [ccc 3&3&-1 -2&-2&1 -4&-5&2] Express the following matrices as the sum of a…
  21. [cc 1&5 -1&2] [cc 3&5 1&-1] Express the following matrices as the sum of a…
  22. If A, B are symmetric matrices of same order, then AB - BA is aA. Skew…
  23. 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
  1. Using elementary transformations, find the inverse of each of the matrices.…
  2. [ll 2&1 1&1] Using elementary transformations, find the inverse of each of the…
  3. [ll 1&3 2&7] Using elementary transformations, find the inverse of each of the…
  4. [ll 2&3 5&7] Using elementary transformations, find the inverse of each of the…
  5. [ll 2&1 7&4] Using elementary transformations, find the inverse of each of the…
  6. [ll 2&5 1&3] Using elementary transformations, find the inverse of each of the…
  7. [ll 3&1 5&2] Using elementary transformations, find the inverse of each of the…
  8. [ll 4&5 3&4] Using elementary transformations, find the inverse of each of the…
  9. [cc 3&10 2&7] Using elementary transformations, find the inverse of each of the…
  10. [cc 3&-1 -4&2] Using elementary transformations, find the inverse of each of…
  11. [ll 2&-6 1&-2] Using elementary transformations, find the inverse of each of…
  12. [cc 6&-3 -2&1] Using elementary transformations, find the inverse of each of…
  13. [cc 2&-1 -3&2] Using elementary transformations, find the inverse of each of…
  14. [ll 2&1 4&2] Using elementary transformations, find the inverse of each of the…
  15. [ccc 2&-3&3 2&2&3 3&-2&2] Using elementary transformations, find the inverse of…
  16. [ccc 1&3&-2 -3&0&-5 2&5&0] Using elementary transformations, find the inverse…
  17. [ccc 2&0&-1 5&1&0 0&1&3] Using elementary transformations, find the inverse of…
  18. Matrices A and B will be inverse of each other only ifA. AB = BA B. AB = BA = 0…
Miscellaneous Exercise
  1. Let, show that (aI + bA)n = anI + nan-1bA, where I is the identity matrix of…
  2. If a = [lll 1&1&1 1&1&1 1&1&1] , prove that .
  3. If a = [ll 3&-4 1&-1] then prove that a^n = [cc 1+2n&-4n n&1-2n] where n is any…
  4. If A and B are symmetric matrices, prove that AB - BA is a skew symmetric…
  5. Show that the matrix B’AB is symmetric or skew symmetric according as A is…
  6. Find the values of x, y, z if the matrix satisfy a = [ccc 0&2y x&-z x&-y] the…
  7. For what values of x: [lll 1&2&1] [lll 1&2&0 2&0&1 1&0&2] [0 2 x] = 0?…
  8. If a = [cc 3&1 -1&2] show that A^2 - 5A + 7I =0.
  9. Find x, if [rrr x&-5&-1] [lll 1&0&2 0&2&1 2&0&3] [x 4 1] = 0
  10. A manufacturer produces three products x, y, z which he sells in two markets.…
  11. Find the matrix X so that x [ccc 1&2&3 4&5&6] = [ccc -7&-8&-9 2&4&6]…
  12. If A and B are square matrices of the same order such that AB = BA, then prove…
  13. If a = [ll alpha & beta gamma & - alpha] is such that A � = I, thenA. 1 +…
  14. If the matrix A is both symmetric and skew symmetric, thenA. A is a diagonal…
  15. 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 a13, a21, a33, a24, a23.


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) a13 = 19, a21 = 35, a33 = -5, a24 = 12, a23 = .


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 = [aij], whose elements are given by:



Answer:

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

aij = 


Therefore,


a11 = 


a12 = 


a21 = 


a22 = 


Therefore, the required matrix is A = 



Question 5.

Construct a 2 × 2 matrix, A = [aij], whose elements are given by:



Answer:

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

aij = 


Therefore,


a11 = 


a12 = 


a21 = 


a22 = 


Therefore, the required matrix is A = .



Question 6.

Construct a 2 × 2 matrix, A = [aij], whose elements are given by:



Answer:

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

aij = 


Therefore,


a11 = 


a12 = 


a21 = 


a22 = 


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,







a32 = 


a13 = 


a23 = 


a33 = 


a14 = 


a24 = 


a34 = 


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 = 

aij = 2i-j, i = 1,2,3 and j = 1,2,3,4


Therefore,


a11 = 2 × 1 - 1 = 2 - 1 = 1


a21 = 2 × 2 - 1 = 4 - 1 = 3


a31 = 2 × 3 - 1 = 6 - 1 = 5


a12 = 2 × 1 - 2 = 2 - 2 = 0


a22 = 2 × 2 - 2 = 4 - 2 = 2


a32 = 2 × 3 - 2 = 6 - 2 = 4


a13 = 2 × 1 - 3 = 2 - 3 = -1


a23 = 2 × 2 - 3 = 4 - 3 = 1


a33 = 2 × 3 - 3 = 6 - 3 = 3


a14 = 2 × 1 - 4 = 2 - 4 = -2


a24 = 2 × 2 - 4 = 4 - 4 = 0


a34 = 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 - y2= 8
⇒ y2 - 6y + 8 = 0
⇒ y2 - 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 = [aij]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 = [aij]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 29 = 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 A2 – 5A + 6I, if 


Answer:

A2 = A.A 



Now, A2 – 5A + 6I 







Question 30.

If, prove that A3 – 6A2 + 7A + 2I = 0


Answer:

A2 = A.A 



Now, A3 = A2. A





Now, A3 – 6A2 + 7A + 2I 






Therefore, A3 – 6A2 + 7A + 2I = 0



Question 31.

If , find k so that A2 = kA – 2I


Answer:

A2 = A.A 



Now, A2 = 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 sin2 θ

cos 2θ = 2 cos2 θ - 1

and sin2θ = 2 sinθ cosθ



As tanθ = sinθ/cosθ