Question

If A=\(\begin{bmatrix}a&b\\c&d\end{bmatrix}\), then which of the following is skew-symmetric?

(a) AA’

(b) A+A’

(c) 2(A+A’)

(d) A-A’

The question was asked during an interview.

Origin of the question is Symmetric and Skew Symmetric Matrices topic in chapter Matrices of Mathematics – Class 12

Answer 1

The correct choice is (c) 2(A+A’)

To explain: Given that, A=\(\begin{bmatrix}a&b\\c&d\end{bmatrix}\)

⇒A’=\(\begin{bmatrix}a&c\\b&d\end{bmatrix}\)

Let B=A-A’=\(\begin{bmatrix}a&b\\c&d\end{bmatrix}\)–\(\begin{bmatrix}a&c\\b&d\end{bmatrix}\)=\(\begin{bmatrix}a-a&b-c\\c-b&d-d\end{bmatrix}\)=\(\begin{bmatrix}0&b-c\\c-b&0\end{bmatrix}\)

B’=\(\begin{bmatrix}0&c-b\\b-c&0\end{bmatrix}\)=B’

Thus, B=A-A’ is a skew – symmetric.