Question

$\mathrm{If}\left(\begin{array}{cc}2& 9\\ 6& 8\end{array}\right]\mathrm{is}\mathrm{expressed}\mathrm{as}\mathrm{the}\mathrm{sum}\mathrm{of}a\mathrm{symmetric}\mathrm{and}a\mathrm{skew}\mathrm{symmetric}\mathrm{matrix},\mathrm{then}\mathrm{those}\mathrm{are}$

• A. $\left(\begin{array}{cc}2& \frac{15}{2}\\ \frac{15}{2}& 8\end{array}\right]\mathrm{and}\left(\begin{array}{cc}0& \frac{3}{2}\\ -\frac{3}{2}& 0\end{array}\right]$
• B. $\left(\begin{array}{cc}4& 5\\ 3& 8\end{array}\right]\mathrm{and}\left(\begin{array}{cc}3& 2\\ 1& 0\end{array}\right]$
• C. None of these

Answer 1

The correct option is A $\left(\begin{array}{cc}2& \frac{15}{2}\\ \frac{15}{2}& 8\end{array}\right]\mathrm{and}\left(\begin{array}{cc}0& \frac{3}{2}\\ -\frac{3}{2}& 0\end{array}\right]$

$\mathrm{Here}A=\left(\begin{array}{cc}2& 9\\ 6& 8\end{array}\right].\mathrm{Hence},\mathrm{the}\mathrm{matrices}\mathrm{will}\mathrm{be}\left(1\right)\mathrm{symmetric}\mathrm{matrix}-\frac{A+A^t}{2}=\left(\begin{array}{cc}2& \frac{15}{2}\\ \frac{15}{2}& 8\end{array}\right]\mathrm{and}\left(2\right)\mathrm{skew}-\mathrm{symmetric}\mathrm{matrix}-\frac{A-A^t}{2}=\left(\begin{array}{cc}0& \frac{3}{2}\\ \frac{-3}{2}& 0\end{array}\right]$