Question

If $\left[\begin{array}{cc}1& 3\\ 2& 1\end{array}\right]$, find the determinant of the matrix $A^2-2A.$

Answer 1

Given

$A=\left[\begin{array}{cc}1& 3\\ 2& 1\end{array}\right]$

To find determinant $A^2-2A$

Now,

$|A^2-2A|=\left|A\right(A-2I\left)\right|$

(Taking A common)

$=\left|A\right||A-2I|(\left|AB\right|=\left|A\right|\left|B\right|)$

$=\left[\begin{array}{cc}1& 3\\ 2& 1\end{array}\right]\times \left[\begin{array}{cc}1& 3\\ 2& 1\end{array}\right]-2\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

$=(1-6)\times |\left[\begin{array}{cc}1& 3\\ 2& 1\end{array}\right]-\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]|$

$-5\times \left|\left[\begin{array}{cc}1-2& 3-0\\ 2-0& 1-2\end{array}\right]\right|$

$=-5\times \left[\begin{array}{cc}-1& 3\\ 2& -1\end{array}\right]$

$=-5\times (-1\times 1-6)$

$=-5\times (1-6)$

$=-5\times -5$

$=25$

$\therefore |A-2A|=25$