Construct the matrix A = [aij]3×3 where aij = i − j. State whether A is symmetric or skew-symmetric.

Exercise 2.2 | Q 10 | Page 47

QUESTION

Construct the matrix A = [a_ij]_3×3 where a_ij = i − j. State whether A is symmetric or skew-symmetric.

SOLUTION

A = [a_ij]_3x3  

∴ A = ${\displaystyle \left[\begin{array}{ccc}{\text{a}}_{11}& {\text{a}}_{12}& {\text{a}}_{13}\\ {\text{a}}_{21}& {\text{a}}_{22}& {\text{a}}_{23}\\ {\text{a}}_{31}& {\text{a}}_{32}& {\text{a}}_{33}\end{array}\right]}$

Given, a_ij = i – j
∴ a₁₁ = 1 – 1 = 0, a₁₂ = 1 – 2 = – 1, a₁₃ = 1 – 3 = – 2
a₂₁ = 2 – 1= 1, a₂₂ = 2 – 2 = 0, a₂₃ = 2 – 3 = – 1,
a₃₁ = 3 – 1 = 2, a₃₂ = 3 - 2 = 1, a₃₃ = 3 – 3 = 0

∴ A = ${\displaystyle \left[\begin{array}{ccc}0& -1& -2\\ 1& 0& -1\\ 2& 1& 0\end{array}\right]}$

∴ A^T = ${\displaystyle \left[\begin{array}{ccc}0& 1& 2\\ -1& 0& 1\\ -2& -1& 0\end{array}\right]}$

= ${\displaystyle -\left[\begin{array}{ccc}0& -1& -2\\ 1& 0& -1\\ 2& 1& 0\end{array}\right]}$

∴ A^T = – A i.e., A = – A^T
∴ A is a skew-symmetric matrix.