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
ANSWERED
asked 2021-02-08
Use elementary matrices to write ? in row reduced echelon form. Use the elementary matrices to find the inverse of
\(A=\begin{bmatrix}1 & 0&-2 \\0& 2&1\\0&0&1 \end{bmatrix}\)

Answers (1)

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 \(A \cdot A^{-1}=\begin{bmatrix}1 & 0&-2 \\0 & 2&1\\0&0&1 \end{bmatrix}\begin{bmatrix}1&0&2 \\0&\frac{1}{2}&-\frac{1}{2}\\0&0&1 \end{bmatrix}=\begin{bmatrix}1&0&0 \\0&1&0\\0&0&1 \end{bmatrix}\)
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