Question

The symmetric part of the matrix A= $\left[\begin{array}{ccc}1& 2& 4\\ 6& 8& 2\\ 2& -2& 7\end{array}\right]$.

• A. $\left[\begin{array}{ccc}1& 4& 3\\ 2& 8& 0\\ 3& 0& 7\end{array}\right]$
• B. $\left[\begin{array}{ccc}1& 4& 3\\ 4& 8& 0\\ 3& 0& 7\end{array}\right]$
• C. $\left[\begin{array}{ccc}0& -2& -1\\ -2& 0& -2\\ -1& -2& 0\end{array}\right]$
• D. $\left[\begin{array}{ccc}0& -2& -1\\ 2& 0& 2\\ -1& 2& 0\end{array}\right]$

Answer 1

The correct option is B $\left[\begin{array}{ccc}1& 4& 3\\ 4& 8& 0\\ 3& 0& 7\end{array}\right]$

Symmetric part of any matrix A is given by $\frac{A+A^{\prime }}{2}$, where $A^{\prime }$ transpose of A

So symmetric part of given matrix $\left[\begin{array}{ccc}1& 2& 4\\ 6& 8& 2\\ 2& -2& 7\end{array}\right]$

is $=\frac{1}{2}($ $\left[\begin{array}{ccc}1& 2& 4\\ 6& 8& 2\\ 2& -2& 7\end{array}\right]$+$\left[\begin{array}{ccc}1& 6& 2\\ 2& 8& -2\\ 4& 2& 7\end{array}\right]$ $)$

$=\frac{1}{2}$$\left[\begin{array}{ccc}2& 8& 6\\ 8& 16& 0\\ 6& 0& 14\end{array}\right]$

$=\left[\begin{array}{ccc}1& 4& 3\\ 4& 8& 0\\ 3& 0& 7\end{array}\right]$