Find the inverse of the following matrix using transformation method.

Engineering

Mathematics

Introduction to Matrix

Algebra of Matrices

Inverse of a Matrix

Question

Find the inverse of the following matrix using transformation method.
$[\begin{array}{cc} 1 & 2 \\ 2 & - 1 \end{array}]$

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Given,

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

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

R₂ → R₂ – 2R₁

[\begin{array}{cc} 1 & 2 \\ 0 & - 5 \end{array} ∣ \begin{array}{cc} 1 & 0 \\ - 2 & 1 \end{array}]

[\begin{array}{cc} 1 & 2 \\ 0 & 1 \end{array} ∣ \begin{array}{cc} 1 & 0 \\ \frac{2}{5} & \frac{- 1}{5} \end{array}]

R₁ + R₁ – 2R₂

[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} ∣ \begin{array}{cc} \frac{1}{5} & \frac{2}{5} \\ \frac{2}{5} & \frac{- 1}{5} \end{array}]

Hence,

A^{- 1} = [\begin{array}{cc} \frac{1}{5} & \frac{2}{5} \\ \frac{2}{5} & \frac{- 1}{5} \end{array}]