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

Engineering

Mathematics

Introduction to Matrix

Algebra of Matrices

Inverse of a Matrix

Question

Using elementary tansormations, find the inverse of each of the matrices, if it exists in
$[\begin{array}{cc} 1 & 3 \\ 2 & 7 \end{array}]$

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$A = [\begin{array}{cc} 1 & 3 \\ 2 & 7 \end{array}]$

|A| = 7 – 6 = 1

Now, A = IA

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

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

[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}] = [\begin{array}{cc} 7 & - 3 \\ - 2 & 1 \end{array}] A (R_{1} \to R_{1} - 3 R_{2})

Hence,

A^{- 1} = [\begin{array}{cc} 7 & - 3 \\ - 2 & 1 \end{array}]