If , then : A

Engineering

Mathematics

Introduction to Matrix

Algebra of Matrices

Inverse of a Matrix

Question

If $A = [\begin{array}{ccc} 3 & - 3 & 4 \\ 2 & - 3 & 4 \\ 0 & - 1 & 1 \end{array}]$ , then : A^–1 =

A⁴

A³

A²

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Given

$A = [\begin{array}{lll} 3 - 3 4 \\ 2 - 3 4 \\ 0 - 1 1 \end{array}]$

$A^{- 1} = \frac{1}{| A |} \cdot adj \cdot A$

|A| = 3(– 3 + 4) + 3(2 – 0) + 4(– 2 – 0)

= 3 + 6 – 8

= 1

$∴ A^{- 1} = \frac{1}{1} [\begin{array}{ccc} 1 & - 2 + 1 & 0 \\ - 2 & 3 & - 4 \\ - 2 & 3 & - 3 \end{array}]$

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

A³ = A · A · A

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

$A^{3} = [\begin{array}{ccc} 1 & - 1 & 0 \\ - 2 & 3 & - 4 \\ - 2 & 3 & - 3 \end{array}] = A^{- 1}$