Find k, if A= [ ll 3 & -2 4 & -2 ] and A^2=k A-21, where I is identity matrix of order 2

A^2=k A-2 l A A+21=k A [ ll 3 & -2 4 & -2 ] [ ll 3 & -2 4 & -2 ]+2 [ ll 1 & 0 0 & 1 ]= k [ ll 3 & -2 4 & -2 ] [ cc 9-8 & -6+4 12-8 & -8+4 ]+ [ cc 2 & 0 0 & 2 ]= [ cc 3 k & -2 k 4 k & -2 k ] [ cc 1 & -2 4 & -4 ]+ [ cc 2 & 0 0 & 2 ]= [ cc 3 k & -2 k 4 k & -2 k ] [ cc 1+2 & -2+0 4+0 & -4+2 ]= [ cc 3 k & -2 k 4 k & -2 k ] [ rr 3 & -2 4 & -2 ]= [ cc 3 k & -2 k 4 k & -2 k ] By equality of matrices, we get 3K = 3 k = 1

Find k, if A= [ ll 3 & -2 4 & -2 ] and A^2=k A-21, where I is ide…

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Question

Find $k$, if $A=\left[\begin{array}{ll}3 & -2 \\ 4 & -2\end{array}\right]$ and $A^2=k A-21$, where $I$ is identity matrix of order $2$

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Answer

$A^2=k A-2 l$
$\therefore A \cdot A+21=k A$
$\therefore\left[\begin{array}{ll}3 & -2 \\4 & -2\end{array}\right]\left[\begin{array}{ll}3 & -2 \\4 & -2\end{array}\right]+2\left[\begin{array}{ll}1 & 0 \\0 & 1\end{array}\right]= k \left[\begin{array}{ll}3 & -2 \\4 & -2\end{array}\right] $
$\therefore\left[\begin{array}{cc}9-8 & -6+4 \\12-8 & -8+4\end{array}\right]+\left[\begin{array}{cc}2 & 0 \\0 & 2\end{array}\right]=\left[\begin{array}{cc}3 k & -2 k \\4 k & -2 k\end{array}\right] $
$\therefore\left[\begin{array}{cc}1 & -2 \\4 & -4\end{array}\right]+\left[\begin{array}{cc}2 & 0 \\0 & 2\end{array}\right]=\left[\begin{array}{cc}3 k & -2 k \\4 k & -2 k\end{array}\right] $
$\therefore\left[\begin{array}{cc}1+2 & -2+0 \\4+0 & -4+2\end{array}\right]=\left[\begin{array}{cc}3 k & -2 k \\4 k & -2 k\end{array}\right] $
$\therefore\left[\begin{array}{rr}3 & -2 \\4 & -2\end{array}\right]=\left[\begin{array}{cc}3 k & -2 k \\4 k & -2 k\end{array}\right] $
$\therefore$ By equality of matrices, we get
$3K = 3$
$\therefore k = 1$