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Question
For what value of x, is the matrix \[A = \begin{bmatrix}0 & 1 & - 2 \\ - 1 & 0 & 3 \\ x & - 3 & 0\end{bmatrix}\] a skew-symmetric matrix?
Solution
Since, A is a skew symmetric matrix
∴ AT = −A
\[\begin{bmatrix}0 & 1 & - 2 \\ - 1 & 0 & 3 \\ x & - 3 & 0\end{bmatrix}^T = - \begin{bmatrix}0 & 1 & - 2 \\ - 1 & 0 & 3 \\ x & - 3 & 0\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}0 & - 1 & x \\ 1 & 0 & - 3 \\ - 2 & 3 & 0\end{bmatrix} = \begin{bmatrix}0 & - 1 & 2 \\ 1 & 0 & - 3 \\ - x & 3 & 0\end{bmatrix}\]
Corresponding elements of equal matrices are equal
\[ \Rightarrow x = 2\]
Hence, the value of x is 2.
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