Question

# Use elementary matrices to write ? in row reduced echelon form. Use the elementary matrices to find the inverse of A=begin{bmatrix}1 & 0&-2 0& 2&10&0&1 end{bmatrix}

Matrices
Use elementary matrices to write ? in row reduced echelon form. Use the elementary matrices to find the inverse of
$$A=\begin{bmatrix}1 & 0&-2 \\0& 2&1\\0&0&1 \end{bmatrix}$$

2021-02-09
Step 1
Let $$A=\begin{bmatrix}1 & 0&-2 \\0& 2&1\\0&0&1 \end{bmatrix}$$
Row reduced echelon form
$$R_1 \rightarrow R_1 +2R_3$$
$$\sim\begin{bmatrix}1 & 0&0 \\0& 2&1\\0&0&1 \end{bmatrix}$$
$$R_2 \rightarrow R_2-R_3$$
$$\sim\begin{bmatrix}1 & 0&0 \\0& 2&0\\0&0&1 \end{bmatrix}$$
Gauss Jordon elimination :to find the matrix
$$[A/I]=\begin{bmatrix}1 & 0&-2&|&1&0&0 \\0& 2&1&|&0&1&0\\0&0&1&|&0&0&1 \end{bmatrix}$$
$$R_2 \rightarrow \frac{R_2}{2} \sim\begin{bmatrix}1 & 0&-2&|&1&0&0 \\0& 1&\frac{1}{2}&|&0&\frac{1}{2}&0\\0&0&1&|&0&0&1 \end{bmatrix}$$
$$R_1 \rightarrow R_1+2R_3$$
$$\sim\begin{bmatrix}1 & 0&0&|&1&0&2 \\0& 1&0&|&0&\frac{1}{2}&0\\0&0&1&|&0&0&1 \end{bmatrix}$$
$$R_2 \rightarrow R_2-\frac{R_3}{2}$$
$$\sim\begin{bmatrix}1 & 0&0&|&1&0&2 \\0& 1&0&|&0&\frac{1}{2}&-\frac{1}{2}\\0&0&1&|&0&0&1 \end{bmatrix}$$
$$\therefore A^{-1}=\begin{bmatrix}1 & 0&2 \\0 & \frac{1}{2}&-\frac{1}{2}\\0&0&1 \end{bmatrix}$$
Varification $$A \cdot A^{-1}=\begin{bmatrix}1 & 0&-2 \\0 & 2&1\\0&0&1 \end{bmatrix}\begin{bmatrix}1&0&2 \\0&\frac{1}{2}&-\frac{1}{2}\\0&0&1 \end{bmatrix}=\begin{bmatrix}1&0&0 \\0&1&0\\0&0&1 \end{bmatrix}$$