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(B) Any square matrix $A$ can be expressed as the sum of a symmetric matrix $B$ and a skew-symmetric matrix $C$, where $B = \frac{1}{2}(A + A^T)$ and

Vedclass · Accepted Answer

$\begin{bmatrix} 0 & 0 & 0 \ 0 & 0 & 0.5 \ 0 & -0.5 & 0 \end{bmatrix}$

Vedclass · Answer

(B) Any square matrix $A$ can be expressed as the sum of a symmetric matrix $B$ and a skew-symmetric matrix $C$, where $B = \frac{1}{2}(A + A^T)$ and $C = \frac{1}{2}(A - A^T)$.Given $A = \begin{bmatrix} 1 & 0 & 1 \ 0 & 2 & 0 \ 1 & -1 & 4 \end{bmatrix}$.Then $A^T = \begin{bmatrix} 1 & 0 & 1 \ 0 & 2 & -1 \ 1 & 0 & 4 \end{bmatrix}$.Now, $C = \frac{1}{2}(A - A^T) = \frac{1}{2} \left( \begin{bmatrix} 1 & 0 & 1 \ 0 & 2 & 0 \ 1 & -1 & 4 \end{bmatrix} - \begin{bmatrix} 1 & 0 & 1 \ 0 & 2 & -1 \ 1 & 0 & 4 \end{bmatrix} ight)$.$C = \frac{1}{2} \begin{bmatrix} 0 & 0 & 0 \ 0 & 0 & 1 \ 0 & -1 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 \ 0 & 0 & 0.5 \ 0 & -0.5 & 0 \end{bmatrix}$.Thus, the correct option is $B$.

If $A = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 2 & 0 \\ 1 & -1 & 4 \end{bmatrix}$, $A = B + C$, $B = B^T$ and $C = -C^T$, then $C = $

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