Question

If [a_ij]_3×3 where a_ij = 2(i – j). Find A and A^T. State whether A and A^T are symmetric or skew-symmetric matrices?

Answer 1

A = [a_ij]_3×3 = ${\displaystyle \left[\begin{array}{ccc}{\text{a}}_{11}& {\text{a}}_{12}& {\text{a}}_{13}\\ {\text{a}}_{21}& {\text{a}}_{22}& {\text{a}}_{23}\\ {\text{a}}_{31}& {\text{a}}_{32}& {\text{a}}_{33}\end{array}\right]}$

Given that : a_ij = 2(i – j),

∴ a₁₁ = 2(1 – 1) = 0,

a₁₂ = 2(1 – 2) = –2,

a₁₃ = 2(1 – 3)= –4,

a₂₁ = 2(2 – 1) = 2,

a₂₂ = 2(2 – 2) = 0,

a₂₃ = 2(2 – 3) = – 2,

a₃₁ = 2(3 – 1) = 4,

a₃₂ = 2(3 – 2) = 2,

a₃₃ = 2(3 – 3) = 0

∴ A = ${\displaystyle \left[\begin{array}{ccc}0& -2& -4\\ 2& 0& -2\\ 4& 2& 0\end{array}\right]}$

∴ A^T = ${\displaystyle \left[\begin{array}{ccc}0& 2& 4\\ -2& 0& 2\\ -4& -2& 0\end{array}\right]}$

∴ A^T = ${\displaystyle -\left[\begin{array}{ccc}0& -2& -4\\ 2& 0& -2\\ 4& 2& 0\end{array}\right]=-\text{A}}$

∴ A^T = – A and A = – A^T

∴ A and A^T both are skew-symmetric matrices.