if matrix A = [ (2 0 1) (2 1 3) (1 -1 0) ] find A2 -5A + 4I. and find

1.	if matrix A = [ (2 0 1) (2 1 3) (1 -1 0) ] find A2 -5A + 4I. and find a matrix X such that A2 -5A +4I + X =0
Answer» A = \(\begin{bmatrix}2&0&1\\2&1&3\\1&-1&0\end{bmatrix}\) \(\therefore\) A² = \(\begin{bmatrix}2&0&1\\2&1&3\\1&-1&0\end{bmatrix}\)\(\begin{bmatrix}2&0&1\\2&1&3\\1&-1&0\end{bmatrix}\) = \(\begin{bmatrix}5&-1&2\\9&-2&5\\0&-1&-2\end{bmatrix}\) A² - 5A + 4I = \(\begin{bmatrix}5&-1&2\\9&-2&5\\0&-1&-2\end{bmatrix}\) - 5\(\begin{bmatrix}2&0&1\\2&1&3\\1&-1&0\end{bmatrix}\) + 4 \(\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}\) = \(\begin{bmatrix}5-10+4&-1+0+0&2-5+0\\9-10+0&-2-5+4&5-15+0\\0-5+0&-1+5+0&-2+0+4\end{bmatrix}\) = \(\begin{bmatrix}-1&-1&-3\\-1&-3&-10\\-5&4&2\end{bmatrix}\) Let A² - 5A + 4I + X = 0 Then X = -(A² - 5A + 4I) = \(-\begin{bmatrix}-1&-1&-3\\-1&-3&-10\\-5&4&2\end{bmatrix}\) = \(\begin{bmatrix}1&1&3\\1&3&10\\5&-4&2\end{bmatrix}\)

if matrix A = [ (2 0 1) (2 1 3) (1 -1 0) ] find A2 -5A + 4I. and find a matrix X such that A2 -5A +4I + X =0

Answer»

A = \(\begin{bmatrix}2&0&1\\2&1&3\\1&-1&0\end{bmatrix}\)

\(\therefore\) A² = \(\begin{bmatrix}2&0&1\\2&1&3\\1&-1&0\end{bmatrix}\)\(\begin{bmatrix}2&0&1\\2&1&3\\1&-1&0\end{bmatrix}\)

= \(\begin{bmatrix}5&-1&2\\9&-2&5\\0&-1&-2\end{bmatrix}\)

A² - 5A + 4I = \(\begin{bmatrix}5&-1&2\\9&-2&5\\0&-1&-2\end{bmatrix}\) - 5\(\begin{bmatrix}2&0&1\\2&1&3\\1&-1&0\end{bmatrix}\) + 4 \(\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}\)

= \(\begin{bmatrix}5-10+4&-1+0+0&2-5+0\\9-10+0&-2-5+4&5-15+0\\0-5+0&-1+5+0&-2+0+4\end{bmatrix}\)

= \(\begin{bmatrix}-1&-1&-3\\-1&-3&-10\\-5&4&2\end{bmatrix}\)

Let A² - 5A + 4I + X = 0

Then X = -(A² - 5A + 4I)

= \(-\begin{bmatrix}-1&-1&-3\\-1&-3&-10\\-5&4&2\end{bmatrix}\)

= \(\begin{bmatrix}1&1&3\\1&3&10\\5&-4&2\end{bmatrix}\)

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