If the matrix A = begin{bmatrix} 4 & 1 3 & 2 | vedclass

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(B) Step $1$: $A$ square matrix $A$ can be written as $A = B + C$, where $B = \frac{1}{2}(A + A^T)$ is symmetric and $C = \frac{1}{2}(A - A^T)$ is ske

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$|A| = |B| + |C|$

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(B) Step $1$: $A$ square matrix $A$ can be written as $A = B + C$, where $B = \frac{1}{2}(A + A^T)$ is symmetric and $C = \frac{1}{2}(A - A^T)$ is skew-symmetric.Step $2$: Calculate $A^T = \begin{bmatrix} 4 & 3 \ 1 & 2 \end{bmatrix}$.Step $3$: $B = \frac{1}{2} \left( \begin{bmatrix} 4 & 1 \ 3 & 2 \end{bmatrix} + \begin{bmatrix} 4 & 3 \ 1 & 2 \end{bmatrix} ight) = \frac{1}{2} \begin{bmatrix} 8 & 4 \ 4 & 4 \end{bmatrix} = \begin{bmatrix} 4 & 2 \ 2 & 2 \end{bmatrix}$.Step $4$: $C = \frac{1}{2} \left( \begin{bmatrix} 4 & 1 \ 3 & 2 \end{bmatrix} - \begin{bmatrix} 4 & 3 \ 1 & 2 \end{bmatrix} ight) = \frac{1}{2} \begin{bmatrix} 0 & -2 \ 2 & 0 \end{bmatrix} = \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix}$.Step $5$: Calculate determinants: $|A| = (4 	imes 2) - (1 	imes 3) = 8 - 3 = 5$. $|B| = (4 	imes 2) - (2 	imes 2) = 8 - 4 = 4$. $|C| = (0 	imes 0) - (-1 	imes 1) = 1$.Step $6$: Check options: $|B| + |C| = 4 + 1 = 5$. Since $|A| = 5$, the relation $|A| = |B| + |C|$ is correct.

If the matrix $A = \begin{bmatrix} 4 & 1 \\ 3 & 2 \end{bmatrix}$ is expressed as the sum of a symmetric matrix $B$ and a skew-symmetric matrix $C$, then which of the following relations is correct?

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