Using elementary tansormations, find the inverse of each of the matrices, if it exists in

Engineering

Mathematics

Introduction to Matrix

Inverse of a Matrix

Introduction to Determinants

Question

Using elementary tansormations, find the inverse of each of the matrices, if it exists in
$[\begin{array}{cc} 2 & 5 \\ 1 & 3 \end{array}]$

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

|A| = 6 – 5 = 1

Hence, A^–1 exists.

Now,

A = IA

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

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

[\begin{array}{cc} 1 & 2 \\ 0 & 1 \end{array}] = [\begin{array}{cc} 1 & - 1 \\ - 1 & 2 \end{array}] A (R_{2} \to R_{2} - R_{1})

[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}] = [\begin{array}{cc} 3 & - 5 \\ - 1 & 2 \end{array}] A (R_{2} \to R_{2} - R_{1})

Hence,

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