Question

If matrix
$P=\left(\begin{array}{ccc}0& 1& -1\\ 4& -3& 4\\ 3& -3& 4\end{array}\right]=Q+R$, where Q is symmetric matrix and R is skew symmetric matrix. Then find matrix Q and R.

• A. $Q=\frac{1}{2}\left[\begin{array}{ccc}1& 5& 2\\ 5& 6& 1\\ 2& 1& 8\end{array}\right],R=\frac{1}{2}\left[\begin{array}{ccc}1& -3& -4\\ 3& 0& 7\\ 4& 5& 0\end{array}\right]$
• B. $Q=\frac{1}{2}\left[\begin{array}{ccc}0& 5& 2\\ 5& 6& 4\\ 2& 1& 8\end{array}\right],R=\frac{1}{2}\left[\begin{array}{ccc}0& -3& -4\\ 3& 0& 7\\ 4& 6& 0\end{array}\right]$
• C. $Q=\frac{1}{2}\left[\begin{array}{ccc}0& 5& 2\\ 5& -6& 1\\ 2& 1& 8\end{array}\right],R=\frac{1}{2}\left[\begin{array}{ccc}0& 3& 4\\ 3& 0& 7\\ 4& -7& 0\end{array}\right]$
• D. $Q=\frac{1}{2}\left[\begin{array}{ccc}0& 5& 2\\ 5& -6& 1\\ 2& 1& 8\end{array}\right],R=\frac{1}{2}\left[\begin{array}{ccc}0& -3& -4\\ 3& 0& 7\\ 4& -7& 0\end{array}\right]$

Answer 1

The correct option is D $Q=\frac{1}{2}\left[\begin{array}{ccc}0& 5& 2\\ 5& -6& 1\\ 2& 1& 8\end{array}\right],R=\frac{1}{2}\left[\begin{array}{ccc}0& -3& -4\\ 3& 0& 7\\ 4& -7& 0\end{array}\right]$

Here matrix P is expressed as the sum of symmetric and skew symmetric matrix. Then
$Q=\frac{1}{2}\left(P+P^T\right)\mathrm{and}R=\frac{1}{2}\left(P-P^T\right)\mathrm{Here},P=\left(\begin{array}{ccc}0& 1& -1\\ 4& -3& 4\\ 3& -3& 4\end{array}\right]\Rightarrow P^T=\left(\begin{array}{ccc}0& 4& 3\\ 1& -3& -3\\ -1& 4& 4\end{array}\right]Q=\frac{1}{2}\left(\left(\begin{array}{ccc}0& 1& -1\\ 4& -3& 4\\ 3& -3& 4\end{array}\right]+\left(\begin{array}{ccc}0& 4& 3\\ 1& -3& -3\\ -1& 4& 4\end{array}\right]\right)=\frac{1}{2}\left(\begin{array}{ccc}0& 5& 2\\ 5& -6& 1\\ 2& 1& 8\end{array}\right]\mathrm{and}R=\frac{1}{2}\left(\left(\begin{array}{ccc}0& 1& -1\\ 4& -3& 4\\ 3& -3& 4\end{array}\right]-\left(\begin{array}{ccc}0& 4& 3\\ 1& -3& -3\\ -1& 4& 4\end{array}\right]\right)=\frac{1}{2}\left(\begin{array}{ccc}0& -3& -4\\ 3& 0& 7\\ 4& -7& 0\end{array}\right]$