Find the inverse of the following matrix using elementary operations.

Engineering

Mathematics

Introduction to Matrix

Algebra of Matrices

Inverse of a Matrix

Question

Find the inverse of the following matrix using elementary operations.

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

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

A = IA

↓

I = BA

Then B = A^–1

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

All off-diagonal = 0

diagonal = 1

Applying R₂ → R₂ + R₁

[\begin{array}{ccc} 1 & 2 & - 2 \\ 0 & 5 & - 2 \\ 0 & - 2 & 1 \end{array}] = [\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}] A

R₁ → R₁ + 2R₃

R₂ → R₂ + 2R₃

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

Now,

R₁ → R₁ + 2R₂

R₃ → R₃ + 2R₂

\underline{[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]} = \begin{array}{ccc} 3 & 2 & 6 \\ 1 & 1 & 2 \\ 2 & 2 & 5 \end{array}] A I B

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