Given a matrix A= (((−1 3 2)),(( 0 1 4)),((−2 3 2)) ) and A^(−1) = (1/(10))(kA+9I−A^2 ). find k.


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Question Number 118231 by bemath last updated on 16/Oct/20

$G i v e n a m a t r i x A = (\begin{matrix} - 1 3 2 \\ 0 1 4 \\ - 2 3 2 \end{matrix})$ $a n d A^{- 1} = \frac{1}{10} (k A + 9 I - A^{2}) .$ $f i n d k .$
	Answered by bobhans last updated on 16/Oct/20

	$\Rightarrow 10 A^{- 1} = k A + 9 I - A^{2}$ $\Rightarrow 10 A^{- 1} . A = {k A}^{2} + 9 A - A^{3}$ $\Rightarrow A^{3} - {k A}^{2} - 9 A + 10 I = 0$ $i t m u s t b e Σ c o e f f i c i e n t = 0$ $\Rightarrow 1 - k - 9 + 10 = 0 \Rightarrow k = 2$ $c h e c k i n g$ $\Rightarrow A^{3} - 2 A^{2} - 9 A + 10 I$ $\Rightarrow A (A^{2} - 2 A - 9 I) + 10 I$ $\Rightarrow A (A (A - 2 I) - 9 I) + 10 I$ $A - 2 I = (\begin{matrix} - 1 3 2 \\ 0 1 4 \\ - 2 3 2 \end{matrix}) - (\begin{matrix} 2 0 0 \\ 0 2 0 \\ 0 0 2 \end{matrix})$ $= (\begin{matrix} - 3 3 2 \\ 0 - 1 4 \\ - 2 3 0 \end{matrix})$ $A (A - 2 I) = (\begin{matrix} - 1 3 2 \\ 0 1 4 \\ - 2 3 2 \end{matrix}) (\begin{matrix} - 3 3 2 \\ 0 - 1 4 \\ - 2 3 0 \end{matrix})$ $= (\begin{matrix} - 1 0 10 \\ - 8 11 4 \\ 2 - 3 8 \end{matrix})$ $n e x t . .$
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