Question

Let $A\left(x\right)=\begin{bmatrix} 0 & x-2 & x-3 \\ x+2 & 0 & x-5 \\ x+3 & x+5 & 0 \end{bmatrix}$ , 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)

$A\left(0\right)=\begin{bmatrix} 0 & -2 & -3 \\ 2 & 0 & -5 \\ 3 & 5 & 0 \end{bmatrix}$
$\left(A \left(0\right)\right)^{T}=\begin{bmatrix} 0 & 2 & 3 \\ -2 & 0 & 5 \\ -3 & -5 & 0 \end{bmatrix}=-A\left(0\right)$
$\Rightarrow A\left(0\right)$ 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, $\left|A \left(0\right) \left(A \left(0\right)\right)^{T}\right|=\left(\left|A \left(0\right)\right|\right)^{2}=0$