If , then show that .

Engineering

Mathematics

Introduction to Matrix

Inverse of a Matrix

Question

If $A = [\begin{matrix} 4 & 5 \\ 2 & 1 \end{matrix}]$ , then show that $A^{- 1} = \frac{1}{6} (A - 5 I)$ .

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A = [\begin{matrix} 4 & 5 \\ 2 & 1 \end{matrix}]

For characterictic equation of A

|A – λl| = 0

[\begin{array}{cc} 4 - λ & 5 \\ 2 & 1 - λ \end{array}] = 0

(4 – λ) (1 – ) – 10 = 0

λ² – 5λ + 4 – 10 = 0

λ² = 5λ – 6 = 0

Acc to cayley Hamiltm theorem, every square matrix satisfied its own characteristic equation

A² – 5A – 6I = 0

– 6I = 5A – A²

Pre multiply by A^–1

– 6A – 1 = (5I – A)

A^{- 1} = \frac{- 1}{6} (5 I - A)

A^{- 1} = \frac{1}{6} (A - 5 I)