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.

Math Word Problem

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.

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 $$A^{-1}=\begin{bmatrix}1 & 0 & 0\\0 & -1& 1 \\0 & -1 & 0\end{bmatrix}$$