Find inverse ?

Engineering

Mathematics

Introduction to Matrix

Inverse of a Matrix

Introduction to Determinants

Question

Find inverse ?
$[\begin{array}{ccc} 2 & - 1 & 4 \\ 4 & 0 & 2 \\ 3 & - 2 & 7 \end{array}]$

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$\to [\begin{array}{ccc} 2 & - 1 & 4 \\ 4 & 0 & 2 \\ 3 & - 2 & 7 \end{array}]$

$∣ \begin{array}{ccc} 2 & - 1 & 4 \\ 4 & 0 & 2 \\ 3 & - 2 & 7 \end{array} ∣ =^{2} ∣ \begin{array}{cc} 0 & 2 \\ - 2 & 7 \end{array} ∣ - (- 1) ∣ \begin{array}{cc} 4 & 2 \\ - 2 & 7 \end{array} ∣ + 4 ∣ \begin{array}{cc} 4 & 0 \\ 3 & - 2 \end{array} ∣$

= 2 (0 – (– 4)) + 1(28 – 6) + 4(– 8)

= 8 + 22 – 32 = – 10

Performing row column operations

(1) R₁ → R₁/2 (2) R₂ → – 4R₁ + R₂

(3) R₃ → – 3R₁ + R³ (4)

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

(5)

R_{3} \to R_{3} + \frac{R_{2}}{2}

(6) R₃ → – 2R₃

(7) R₂ → R₂ + 3R₃ (8) R₁ → R₁ – 2R₃

(9)

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

After performing above operations in the given sequence, we obtain

[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]