Question

If $(A-2I)(A-3I)=0$, where $A=\left[\begin{array}{cc}4& 2\\ -1& x\end{array}\right]$ and $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ , find the value of $x$.

Answer 1

$(A-2I)(A-3I)=0$

$\{\left[\begin{array}{cc}4& 2\\ -1& x\end{array}\right]-\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]\}$ $\{\left[\begin{array}{cc}4& 2\\ -1& x\end{array}\right]-\left[\begin{array}{cc}3& 0\\ 0& 3\end{array}\right]\}$ $=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

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

$\Rightarrow \left[\begin{array}{cc}0& 2x-2\\ -x+1& x^2-5x+4\end{array}\right]$ $=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

Equating corresponding elements of equal matrices.

$-x+1=0$ $2x-2=0$ $x^2-5x+4=0$

$x=1$ $x=1$ $x=1,4$