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

Engineering

Mathematics

Multiplication of Matrices

Symmetric and Skew Symmetric Matrices

Question

Find the inverse of the following martrices by using transformation method.
$[\begin{matrix} 2 & 0 & - 1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{matrix}]$

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

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

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

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

R₂ → R₂ + 2R₁

R₁ → R₁ – R₃

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

R₁ → R₁ + R₂

R₃ → R₃ – 3R₂

$[\begin{matrix} 1 & 0 & 0 & ⋮ & 3 & - 1 & 1 \\ 0 & 0 & 1 & ⋮ & 5 & - 2 & 2 \\ 0 & 1 & 0 & ⋮ & - 15 & 6 & - 5 \end{matrix}] R_{2} \to R_{3}$

$[\begin{matrix} 1 & 0 & 0 & ⋮ & 3 & - 1 & 1 \\ 0 & 1 & 0 & ⋮ & - 15 & 6 & - 5 \\ 0 & 0 & 1 & ⋮ & 5 & - 2 & 2 \end{matrix}]$

$∴ A^{- 1} = [\begin{array}{ccc} 3 & - 1 & 1 \\ - 15 & 6 & - 5 \\ 5 & - 2 & 2 \end{array}]$

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