Find the inverse of the following matrix, using elementary transformations :

Engineering

Mathematics

Introduction to Matrix

Algebra of Matrices

Inverse of a Matrix

Question

Find the inverse of the following matrix, using elementary transformations :
$A = [\begin{array}{ccc} 2 & 3 & 1 \\ 2 & 4 & 1 \\ 3 & 7 & 2 \end{array}]$

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A = IA

[\begin{array}{ccc} 2 & 3 & 1 \\ 2 & 4 & 1 \\ 3 & 7 & 2 \end{array}] = [\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}] . A

R₂ → R₂ – R₁

R₃ → 2R₃ – 3R₁

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

R₁ → R₁ – R₃

R₃ → R₃ – 3R₂

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

R₁ → R₁ / 2

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

So, A^–1

[\begin{array}{ccc} 2 & 0 & - 1 \\ - 1 & 1 & 0 \\ 0 & 0 & 2 \end{array}]