Question

Let $A\left(x\right)=\left[\begin{array}{ccc}0& x-2& x-3\\ x+2& 0& x-5\\ x+3& x+5& 0\end{array}\right]$ , then the matrix $A\left(0\right){\left(A\left(0\right)\right)}^T$ is a

• A. null matrix
• B. symmetric matrix
• C. skew symmetric matrix
• D. non singular matrix

Answer 1

Answer: (B)

$A\left(0\right)=\left[\begin{array}{ccc}0& -2& -3\\ 2& 0& -5\\ 3& 5& 0\end{array}\right]$
${\left(A\left(0\right)\right)}^T=\left[\begin{array}{ccc}0& 2& 3\\ -2& 0& 5\\ -3& -5& 0\end{array}\right]=-A\left(0\right)$
$\Rightarrow A\left(0\right)$ is a skew-symmetric matrix
${\left(A\left(0\right){\left(A\left(0\right)\right)}^T\right)}^T={\left({\left(A\left(0\right)\right)}^T\right)}^T{\left(A\left(0\right)\right)}^T=A\left(0\right){\left(A\left(0\right)\right)}^T$
$\Rightarrow A\left(0\right){\left(A\left(0\right)\right)}^T$ is a symmetric matrix
Also, $\left|A\left(0\right){\left(A\left(0\right)\right)}^T\right|={\left(\left|A\left(0\right)\right|\right)}^2=0$