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  4. If A=[1&2 -1&3] and if A6=K A-205 I then

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Q. If $A=\begin{bmatrix}1&2\\ -1&3\end{bmatrix}$ and if $A^{6}=K A-205 I$ then

Matrices

Solution:

$ A=\begin{bmatrix}1&2\\ -1&3\end{bmatrix}$
$ \Rightarrow A^{2} =\begin{bmatrix}1&2\\ -1&3\end{bmatrix}\begin{bmatrix}1&2\\ -1&3\end{bmatrix}=$
$\begin{bmatrix}-1&8\\ -4&7\end{bmatrix}$
$=4A -5I$
$ \Rightarrow A^{3} =A A^{2}=A(4 A-5 I)$
$=4 A^{2}-5 A $
$=4(4 A-5 I)-5 A $
$=11 A-20 I $
$A^{6} =A^{3} \cdot A^{3}=(11 A-20 I)^{2} $
$=121 A^{2}-440 A+400 I $
$=121(4 A-5 I)-440 A+400 I $
$=44 A-205 I $