Question

What is the symmetric part of 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. $\left[\begin{array}{ccc}1& 4& 3\\ 4& 8& 0\\ 3& 0& 7\end{array}\right]$
C. $\left[\begin{array}{ccc}1& 4& -3\\ 2& 8& 0\\ 3& 0& 7\end{array}\right]$
D. $\left[\begin{array}{ccc}0& -2& 1\\ 2& 0& 2\\ -1& 2& 0\end{array}\right]$

Answer 1

Hint: Using the fact that any matrix A can be written as the sum of a symmetric and a skew symmetric matrix we can solve the given problem.

Formula Used:
If $A$ be a matrix and $A^{\prime }$ be its transpose then $A$ can be decomposed in two parts as
$A={\displaystyle \frac{1}{2}}\left(A+A^{\prime }\right)+{\displaystyle \frac{1}{2}}\left(A-A^{\prime }\right)$
Where,
Symmetric part of matrix = ${\displaystyle \frac{1}{2}}\left(A+A^{\prime }\right)$
Skew symmetric part of matrix =${\displaystyle \frac{1}{2}}\left(A-A^{\prime }\right)$

Complete step by step solution:
Given -$A=\left[\begin{array}{ccc}1& 2& 4\\ 6& 8& 2\\ 2& -2& 7\end{array}\right]$
We know that if $A$ be a matrix and$A^{\prime }$be its transpose then the symmetric part of matrix is given by Symmetric part of matrix = ${\displaystyle \frac{1}{2}}\left(A+A^{\prime }\right)$
Since $A=\left[\begin{array}{ccc}1& 2& 4\\ 6& 8& 2\\ 2& -2& 7\end{array}\right]$
So, $A^{\prime }=\left[\begin{array}{ccc}1& 6& 2\\ 2& 8& -2\\ 4& 2& 7\end{array}\right]$
Now,
${\displaystyle \frac{1}{2}}\left(A+A^{\prime }\right)={\displaystyle \frac{1}{2}}\left[\begin{array}{ccc}1& 2& 4\\ 6& 8& 2\\ 2& -2& 7\end{array}\right]+\left[\begin{array}{ccc}1& 6& 2\\ 2& 8& -2\\ 4& 2& 7\end{array}\right]$
$\Rightarrow {\displaystyle \frac{1}{2}}\left(A+A^{\prime }\right)={\displaystyle \frac{1}{2}}\left[\begin{array}{ccc}2& 8& 6\\ 8& 16& 0\\ 6& 0& 14\end{array}\right]$
$\Rightarrow {\displaystyle \frac{1}{2}}\left(A+A^{\prime }\right)=\left[\begin{array}{ccc}1& 4& 3\\ 4& 8& 0\\ 3& 0& 7\end{array}\right]$
∴ Symmetric part =$\left[\begin{array}{ccc}1& 4& 3\\ 4& 8& 0\\ 3& 0& 7\end{array}\right]$

Option ‘B’ is correct

Note: Students may get confused in formula and wrong application of formula will give wrong answer. Correct formulas are –
Symmetric part = ${\displaystyle \frac{1}{2}}\left(A+A^{\prime }\right)$
Skew symmetric =${\displaystyle \frac{1}{2}}\left(A-A^{\prime }\right)$