If A is a symmetric matrix and B is a skew-symmetric matrix such that A+B=pmatrix2&35&-1pm…

Question

If $A$ is a symmetric matrix and $B$ is a skew-symmetric matrix such that $A+B=\begin{pmatrix}2&3\5&-1\end{pmatrix}$, then $AB$ is equal to:

Accepted Answer

Let $A=\begin{pmatrix}a&c\c&b\end{pmatrix}$, $B=\begin{pmatrix}0&d\-d&0\end{pmatrix}$. Then $A+B=\begin{pmatrix}a&c+d\c-d&b\end{pmatrix}=\begin{pmatrix}2&3\5&-1\end{pmatrix}$.

So $a=2,\ b=-1,\ c-d=5,\ c+d=3$, giving $c=4,\ d=-1$.

$A=\begin{pmatrix}2&4\4&-1\end{pmatrix}$, $B=\begin{pmatrix}0&-1\1&0\end{pmatrix}$.

$AB=\begin{pmatrix}2&4\4&-1\end{pmatrix}\begin{pmatrix}0&-1\1&0\end{pmatrix}=\begin{pmatrix}4&-2\-1&-4\end{pmatrix}$.

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