Question

If A=
$\binom{23}{10}$
=P+Q where P is symmetric and Q is skew symmetric martix, then find the matrix Q​

Answer 1

Given:- $A=\left[\begin{array}{ll}2 & 3 \\1 & 0\end{array}\right]_{2 \times 2}$

$\text { P is syrametric }\\P=\frac{1}{2}\left(A+A^{+}\right)\\Q \text { is skew symmetric }\\Q=\frac{1}{2}\left(A-A^{\top}\right)\\A^{\top}=\left[\begin{array}{ll}2 & 1 \\3 & 0\end{array}\right]$

NOw as we know that ,$Q=\frac{1}{2}\left(A-A^{T}\right)$

Then,

$Q=\frac{1}{2}\left(\left[\begin{array}{ll}2 & 3 \\1 & 0\end{array}\right]-\left[\begin{array}{ll}2 & 1 \\3 & 0\end{array}\right]\right)$

$=\left(\frac{1}{2}\left[\begin{array}{cc}0 & 2 \\-2 & 0\end{array}\right]\right.$

Q =  $\left[\begin{array}{ll}0 & 1 \\-1 & 0\end{array}\right]$