If , show that A 2 – 5A + 7I = O. Hence find A –1 .

Question

Philoid · Accepted Answer

We have A² = A.A = .

So A² – 5A + 7I =

Hence A² – 5A + 7I = 0

∴ A.A – 5A = -7I

Now post multiply with A^-1

So A.A.A^-1-5A.A^-1 = -7I.A^-1

→ A.I – 5I = -7I.A^-1 {since A.A^-1 = I}

A – 5I = -7A^-1 {since X.I = X}

If , show that A2 – 5A + 7I = O. Hence find A–1.