Find the inverse of the following martrices by using transformation method.

Engineering

Mathematics

Introduction to Matrix

Algebra of Matrices

Inverse of a Matrix

Question

Find the inverse of the following martrices by using transformation method.
$[\begin{array}{ccc} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \end{array}]$

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$[\begin{matrix} 0 & 1 & 2 & ⋮ & 1 & 0 & 0 \\ 1 & 2 & 3 & ⋮ & 0 & 1 & 0 \\ 3 & 1 & 1 & ⋮ & 0 & 0 & 1 \end{matrix}] R_{2} \leftrightarrow R_{1}$

$[\begin{matrix} 1 & 2 & 3 & ⋮ & 0 & 1 & 0 \\ 0 & 1 & 2 & ⋮ & 1 & 0 & 0 \\ 3 & 1 & 1 & ⋮ & 0 & 0 & 1 \end{matrix}] R_{1} \to R_{3} - 3 R_{1}$

$[\begin{matrix} 1 & 2 & 3 & ⋮ & 0 & 1 & 0 \\ 0 & 1 & 2 & ⋮ & 1 & 0 & 0 \\ 0 & - 5 & - 8 & ⋮ & 0 & - 3 & 1 \end{matrix}] R_{3} \to R_{3} + 5 R_{2}$

$[\begin{matrix} 1 & 2 & 3 & ⋮ & 0 & 1 & 0 \\ 0 & 1 & 2 & ⋮ & 1 & 0 & 0 \\ 0 & 0 & 2 & ⋮ & 5 & - 3 & 1 \end{matrix}] \begin{matrix} R_{2} \to R_{2} - R_{3} \\ R_{1} \to R_{1} - 2 R_{2} \end{matrix}$

$[\begin{matrix} 1 & 0 & - 1 & ⋮ & - 2 & 1 & 0 \\ 0 & 1 & 0 & ⋮ & - 4 & 3 & - 1 \\ 0 & 0 & 2 & ⋮ & 5 & - 3 & 1 \end{matrix}] R_{3} \to R_{3} / 2$

$[\begin{matrix} 1 & 0 & - 1 & ⋮ & - 2 & - 2 & 10 \\ 0 & 1 & 0 & ⋮ & - 4 & 3 & - 1 \\ 0 & 0 & 1 & ⋮ & 5 / 2 & - 3 / 2 & 1 / 2 \end{matrix}] R_{1} \to R_{1} + R_{3}$

$[\begin{matrix} 1 & 0 & 0 & ⋮ & - 2 + 5 / 2 & 1 + - 3 / 2 & 1 / 2 \\ 0 & 1 & 0 & ⋮ & - 4 & 3 & - 1 \\ 0 & 0 & 1 & ⋮ & 5 / 2 & - 3 / 2 & 1 / 2 \end{matrix}]$

$∴ A^{- 1} = [\begin{array}{ccc} 1 / 2 & - 1 / 2 & 1 / 2 \\ - 4 & 3 & - 1 \\ 5 / 2 & - 3 / 2 & 1 / 2 \end{array}]$