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Mathematics Question on Matrices

if A=[3−4 1−1]A = \begin{bmatrix}3&-4\\\ 1&-1\end{bmatrix} is the sum of a symmetric matrix BB and a skew-symmetric matrix CC, then CC is

A

[1−52 520]\begin{bmatrix}1&-\frac{5}{2}\\\ \frac{5}{2}&0\end{bmatrix}

B

[1−52 521]\begin{bmatrix}1&-\frac{5}{2}\\\ \frac{5}{2}&1\end{bmatrix}

C

[0−52 520]\begin{bmatrix}0&-\frac{5}{2}\\\ \frac{5}{2}&0\end{bmatrix}

D

[0−32 521]\begin{bmatrix}0&-\frac{3}{2}\\\ \frac{5}{2}&1\end{bmatrix}

Answer

[0−52 520]\begin{bmatrix}0&-\frac{5}{2}\\\ \frac{5}{2}&0\end{bmatrix}

Explanation

Solution

A=[3−4 1−1]A = \begin{bmatrix}3&-4\\\ 1&-1\end{bmatrix} A=(A+A′2)+(A−A′2)=B+CA= \left(\frac{A + A'}{2}\right)+\left(\frac{A-A'}{2}\right) = B + C [where BB and CC are symmetric and skew-symmetric matrices respectively] Now, C = \frac{A - A'}{2} = \frac{1}{2 } \left\\{\begin{bmatrix}3&-4\\\ 1&-1\end{bmatrix} - \begin{bmatrix}3&1\\\ -4&-1\end{bmatrix}\right\\} =12[0−5 50]=[0−52 520] = \frac{1}{2}\begin{bmatrix}0&-5\\\ 5&0\end{bmatrix} = \begin{bmatrix}0&-\frac{5}{2}\\\ \frac{5}{2}&0\end{bmatrix}