Question

Work each problem. Let A=\begin{bmatrix}1 & 0 & 0\\0 & 0& -1 \\0 & 1 & -1\end{bmatrix}. Show that A^{3} {i}_{3}, and use this result to find the inverse of A.

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asked 2021-07-03

Work each problem. Let \(A=\begin{bmatrix}1 & 0 & 0\\0 & 0& -1 \\0 & 1 & -1\end{bmatrix}\). Show that \(A^{3} {i}_{3}\), and use this result to find the inverse of A.

Expert Answers (1)

2021-07-04

We have the matrix
\(\begin{bmatrix}1 & 0 & 0\\0 & 0& -1 \\0 & 1 & -1\end{bmatrix}\)
We need to show that \(A^{3} {i}_{3}\)
First find \(A\times A\)
\(A\times A=\begin{bmatrix}1 & 0 & 0\\0 & 0& -1 \\0 & 1 & -1\end{bmatrix}\times\begin{bmatrix}1 & 0 & 0\\0 & 0& -1 \\0 & 1 & -1\end{bmatrix}\)
\(A^{2}=\begin{bmatrix}1 & 0 & 0\\0 & -1& 1 \\0 & -1 & 0\end{bmatrix}\)
\(A^{3}=A^{2}\times A \begin{bmatrix}1 & 0 & 0\\0 & -1& 1 \\0 & -1 & 0\end{bmatrix}\times \begin{bmatrix}1 & 0 & 0\\0 & 0& -1 \\0 & 1 & -1\end{bmatrix}=\begin{bmatrix}1 & 0 & 0\\0 & 1& 0 \\0 & 0 & 1\end{bmatrix}=I_{3}\)
\(\displaystyle\forall^{{-{1}}}={I}_{{{3}}}\)
\(\displaystyle\forall^{{-{1}}}={A}^{{{3}}}\)
\(\displaystyle\forall^{{-{1}}}={A}^{{{2}}}{A}\)
\(\displaystyle{A}^{{-{1}}}={A}^{{{2}}}\)
Result \(A^{-1}=\begin{bmatrix}1 & 0 & 0\\0 & -1& 1 \\0 & -1 & 0\end{bmatrix}\)

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