If A is symmetric and B is skew symmetric matrix and \(A+B=\begin{bmatrix}1 & -1\\3 & 2\end{bmatrix}\), then matrix A = _______.
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\begin{bmatrix}1 & 1\\1 & 2\end{bmatrix}\)
Let \(M = A + B\). Taking the transpose, with \(A^T = A\) (symmetric) and \(B^T = -B\) (skew symmetric):
\(M^T = A - B\)
Adding the two equations gives the symmetric part of M:
\( \displaystyle A = \frac{1}{2}\left(M + M^T\right) \)
\( \displaystyle = \frac{1}{2}\left(\begin{bmatrix}1 & -1\\3 & 2\end{bmatrix} + \begin{bmatrix}1 & 3\\-1 & 2\end{bmatrix}\right) \)
\( \displaystyle = \frac{1}{2}\begin{bmatrix}2 & 2\\2 & 4\end{bmatrix} \)
\( \displaystyle = \begin{bmatrix}1 & 1\\1 & 2\end{bmatrix} \)
Check: \(B = M - A = \begin{bmatrix}0 & -2\\2 & 0\end{bmatrix}\), which is skew symmetric.