Question

If $A=\left[\begin{array}{c}43\\ 25\end{array}\right],$ find $x+y$ such that $A^2-xA+yI=O.$

Answer 1

Given,

$A=\left[\begin{array}{c}43\\ 25\end{array}\right]$ and $A^2-xA+yI=0$

Therefore,

$\left[\begin{array}{c}43\\ 25\end{array}\right]$ $\left[\begin{array}{c}43\\ 25\end{array}\right]$ $-x$ $\left[\begin{array}{c}43\\ 25\end{array}\right]$ $+y$ $\left[\begin{array}{c}10\\ 01\end{array}\right]=O$

$\Rightarrow \left[\begin{array}{c}2227\\ 1831\end{array}\right]$ $-$ $\left[\begin{array}{c}4x3x\\ 2x5x\end{array}\right]$ $+y$ $\left[\begin{array}{c}10\\ 01\end{array}\right]=O$

$\Rightarrow \left[\begin{array}{c}22-4x+y27-3x+0\\ 18-2x+031-5x+y\end{array}\right]$$=\left[\begin{array}{c}00\\ 00\end{array}\right]$

$\Rightarrow 22-4x+y=0$

$\Rightarrow 4x-y=22$------(1)

$\Rightarrow 18-2x=0$

$\Rightarrow x=9$

Put in (1), $4\times 9-y=22$

$\Rightarrow 36-22=y$

$\Rightarrow y=14$

$\therefore x+y=23$