Using elementary transformations, find the inverse of each of the matrices, if it exists.

Engineering

Mathematics

Introduction to Matrix

Algebra of Matrices

Inverse of a Matrix

Question

Using elementary transformations, find the inverse of each of the matrices, if it exists.

[\begin{array}{cc} 1 & - 1 \\ 2 & 3 \end{array}]

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A = IA [row transformation]

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

R₂ → R₂ – 2R₁

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

R - 2 \to \frac{1}{5} R_{2}

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

R₁ → R₁ + 2R₂

[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}] = [\begin{array}{cc} 3 / 5 & 1 / 5 \\ - 2 / 5 & 1 / 5 \end{array}] A

∴ A^{- 1} = [\begin{array}{cc} 3 / 5 & 1 / 5 \\ - 2 / 5 & 1 / 5 \end{array}]