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

Answer 1

The correct option is D

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

Explanation for the correct option:

Given matrix is, $A=\left[\begin{array}{ccc}1& 2& 4\\ 6& 8& 2\\ 2& -2& 7\end{array}\right]$

All square matrices can be expressed as the sum of a symmetric and skew-symmetric matrix. It is given as,

$A=\frac{A+A^{\text{'}}}{2}+\frac{A-A^{\text{'}}}{2}$.

The first term in the $\mathrm{RHS}$ of this equation is called the symmetric part of the matrix.

Thus, the symmetric part of the given matrix is,

$\begin{array}{cl}& \frac{1}{2}\left(\left[\begin{array}{ccc}1& 2& 4\\ 6& 8& 2\\ 2& -2& 7\end{array}\right]+\left[\begin{array}{ccc}1& 6& 2\\ 2& 8& -2\\ 4& 2& 7\end{array}\right]\right)\\ =& \frac{1}{2}\left[\begin{array}{ccc}2& 8& 6\\ 8& 16& 0\\ 6& 0& 14\end{array}\right]\\ =& \left[\begin{array}{ccc}1& 4& 3\\ 4& 8& 0\\ 3& 0& 7\end{array}\right]\end{array}$

Hence, $\mathrm{Option}\left(\mathrm{D}\right)$ is correct.