Question

Matrix A is expressed as A = B + C where B and C are respectively symmetric & skew symmetric matrix then matrix B is

Given $A=\left[\begin{array}{ccc}1& 3& 2\\ 2& 4& 1\\ 3& 1& 5\end{array}\right]$

• A. $\left[\begin{array}{ccc}1& 5/2& 5/2\\ 5& 8& 2\\ 5& 2& 10\end{array}\right]$
• B. $\left[\begin{array}{ccc}1& 5/2& 5/2\\ 5/2& 4& 1\\ 5/2& 1& 5\end{array}\right]$
• C. $\left[\begin{array}{ccc}2& 5& 5\\ 5& 8& 2\\ 5& 2& 10\end{array}\right]$
• D. $\left[\begin{array}{ccc}1& 5/2& 5/2\\ 5/2& 4& 1\\ 5/2& 1& 10\end{array}\right]$

Answer 1

The correct option is B $\left[\begin{array}{ccc}1& 5/2& 5/2\\ 5/2& 4& 1\\ 5/2& 1& 5\end{array}\right]$

$A=\frac{A+A^T}{2}+\frac{A-A^T}{2}$

Where, $B=\frac{A+A^T}{2}$ is symmetric matrix

$C=\frac{A-A^T}{2}$ is skew symmetric matrix

Now., $A^T=\left[\begin{array}{ccc}1& 2& 3\\ 3& 4& 1\\ 2& 1& 5\end{array}\right]$
$\Rightarrow B=\frac{A+A^T}{2}=\frac{1}{2}\left[\begin{array}{ccc}2& 5& 5\\ 5& 8& 2\\ 5& 2& 10\end{array}\right]$

$=\left[\begin{array}{ccc}1& 5/2& 5/2\\ 5/2& 4& 1\\ 5/2& 1& 5\end{array}\right]$