Using elementary transformations, find the inverse of each of the matrices,

Engineering

Mathematics

Algebra of Matrices

Inverse of a Matrix

Question

Using elementary transformations, find the inverse of each of the matrices,
$[\begin{array}{cc} 3 & - 1 \\ - 4 & 2 \end{array}]$

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A = IA [ row transformation ]
$[\begin{array}{cc} 3 & 10 \\ 2 & 4 = 7 \end{array}] = [\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}] A$

R₁ ↔ 1/3R₁
$[\begin{array}{cc} 1 & 10 / 3 \\ 2 & 7 \end{array}] = [\begin{array}{cc} 1 / 3 & 0 \\ 0 & 1 \end{array}] A$
R₂ → R₂ – 2R₁

$[\begin{array}{cc} 1 & 10 / 3 \\ 0 & 1 / 3 \end{array}] = [\begin{array}{cc} 1 / 3 & 0 \\ - 2 / 3 & 1 \end{array}] A$

R₂ → 3R₂

$[\begin{array}{cc} 1 & 10 / 3 \\ 0 & 1 \end{array}] = [\begin{array}{cc} 1 / 3 & 0 \\ - 2 & 3 \end{array}] A$

R₁ → R₁ – 10 / 3R₂

$[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}] = [\begin{array}{cc} 7 & - 10 \\ - 2 & 3 \end{array}] A$

$∴ A^{- 1} = [\begin{array}{cc} 7 & - 10 \\ - 2 & 3 \end{array}]$