Find the inverse of the following matrix by using elementary row transformation:

Engineering

Mathematics

Multiplication of Matrices

Symmetric and Skew Symmetric Matrices

Question

Find the inverse of the following matrix by using elementary row transformation:
$[\begin{array}{ccc} 3 & - 3 & 4 \\ 2 & - 3 & 4 \\ 0 & - 1 & 1 \end{array}]$

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A = IA

$[\begin{array}{lll} 3 - 3 4 \\ 2 - 3 4 \\ 0 - 1 1 \end{array}] = [\begin{array}{lll} 1 0 0 \\ 0 1 0 \\ 0 0 1 \end{array}] A$

R₁ → R₁ – R₂

R₂ → R₂ – 4R₃

$[\begin{array}{ccc} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 0 & - 1 & 1 \end{array}] = [\begin{array}{ccc} 1 & - 1 & 0 \\ 0 & 1 & - 4 \\ 0 & 0 & 1 \end{array}] A$

R₂ → R₂ – 2R₁

$[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & - 1 & 1 \end{array}] = [\begin{array}{ccc} 1 & - 1 & 0 \\ - 2 & 3 & - 4 \\ 0 & 0 & 1 \end{array}] A$

R₃ → R₂ + R₃

$[\begin{array}{lll} 1 0 0 \\ 0 1 0 \\ 0 0 1 \end{array}] = [\begin{array}{ccc} 1 & - 1 & 0 \\ - 2 & 3 & - 4 \\ - 2 & 3 & - 3 \end{array}] A$

$∴ A^{- 1} = [\begin{array}{ccc} 1 & - 1 & 0 \\ - 2 & 3 & - 4 \\ - 2 & 3 & - 3 \end{array}]$

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