Find the inverse of using elementary transformations.

Engineering

Mathematics

Introduction to Matrix

Algebra of Matrices

Inverse of a Matrix

Question

Find the inverse of $A = [\begin{array}{cc} 3 & - 1 \\ - 4 & 1 \end{array}]$ using elementary transformations.

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Given,

A = [\begin{array}{cc} 3 & - 1 \\ - 4 & 1 \end{array}]

For elementary row operations, we write

A = IA

\Rightarrow [\begin{array}{cc} 3 & - 1 \\ - 4 & 1 \end{array}] = [\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}] A

[Applying

R_{1} \to R_{1} + \frac{1}{2} R_{2}

]

[\begin{array}{cc} 1 & - \frac{1}{2} \\ - 4 & 1 \end{array}] = [\begin{array}{cc} 1 & \frac{1}{2} \\ 0 & 1 \end{array}] A

[Applying R₂ → R₂ + 4R₁]

[\begin{array}{cc} 1 & - 1 / 2 \\ 0 & - 1 \end{array}] = [\begin{array}{cc} 1 & 1 / 2 \\ 4 & 3 \end{array}] A

[Applying R₂ → – R₂]

[\begin{array}{cc} 1 & - 1 / 2 \\ 0 & 1 \end{array}] = [\begin{array}{cc} 1 & 1 / 2 \\ - 4 & - 3 \end{array}] A

[Applying

R_{1} \to R_{1} + \frac{1}{2} R_{2}

]

[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}] = [\begin{array}{cc} - 1 & - 1 \\ - 4 & - 3 \end{array}] A

I = A^–1A

\Rightarrow A^{- 1} = [\begin{array}{cc} - 1 & - 1 \\ - 4 & - 3 \end{array}]