if M = [ cc 4 & 1 -1 & 2 ] show that 6 m-m^2=9 I where I is 2 2 u… - Vidyadip

Question

if $M =\left[\begin{array}{cc}4 & 1 \\ -1 & 2\end{array}\right]$ show that $6 m-m^2=9 I$ where $I$ is $2 \times 2$ unit matrix.

✓

Answer

$M^2=\left[\begin{array}{cc}4 & 1 \\ -1 & 2\end{array}\right]\left[\begin{array}{cc}4 & 1 \\ -1 & 2\end{array}\right]$
$=\left[\begin{array}{cc}16-1 & 4+2 \\ -4-2 & -1+4\end{array}\right]=\left[\begin{array}{cc}15 & 6 \\ -6 & 3\end{array}\right]$
$\begin{array}{l}6 M-M^2=6\left[\begin{array}{cc}4 & 1 \\ -1 & 2\end{array}\right]-\left[\begin{array}{cc}15 & 6 \\ -6 & 3\end{array}\right] \end{array}$
$=\left[\begin{array}{cc}24 & 6 \\ -6 & 12\end{array}\right]-\left[\begin{array}{cc}15 & 6 \\ -6 & 3\end{array}\right] $
$ =\left[\begin{array}{ll}9 & 0 \\ 0 & 9\end{array}\right] $
$ =9\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
$=9 $
hence proved

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