If A is a symmetric matrix and B is a skew-symmetric matrix such that A + B = [[ pmatrix 2…

Question

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

Accepted Answer

$$A + B = \left[ {\begin{pmatrix} 2 & 3  \
   5 & { - 1}  \

\end{pmatrix} } \right] = P(say)$$

Now $$A = {{P + {P^T}} \over 2}\& B = {{P - {P^T}} \over 2}$$

So $$A = {1 \over 2}\left( {\left[ {\begin{pmatrix} 2 & 3  \
   5 & { - 1}  \

\end{pmatrix} } \right] + \left[ {\begin{pmatrix} 2 & 5  \
   3 & { - 1}  \

\end{pmatrix} } \right]} \right) = \left[ {\begin{pmatrix} 2 & 4  \
   4 & { - 1}  \

\end{pmatrix} } \right]$$

$$B = {1 \over 2}\left( {\left[ {\begin{pmatrix} 2 & 3  \
   5 & { - 1}  \

\end{pmatrix} } \right] - \left[ {\begin{pmatrix} 2 & 5  \
   3 & { - 1}  \

\end{pmatrix} } \right]} \right) = \left[ {\begin{pmatrix} 0 & { - 1}  \
   1 & 0  \

\end{pmatrix} } \right]$$

So $$AB = \left( {\left[ {\begin{pmatrix} 2 & 4  \
   4 & { - 1}  \

\end{pmatrix} } \right]\left[ {\begin{pmatrix} 0 & { - 1}  \
   1 & 0  \

\end{pmatrix} } \right]} \right) = \left[ {\begin{pmatrix} 4 & { - 2}  \
   { - 1} & { - 4}  \

\end{pmatrix} } \right]$$

Solutions · 1