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

Q13 of 127 Page 122

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

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}

Find the inverse of each of the matrices (if it exists)

Let . Verify that (AB)^–1 = B^–1 A^–1.

For the matrix , find the numbers a and b such that A² + aA + bI = O.

For the matrix

Show that A³– 6A² + 5A + 11 I = O. Hence, find A^–1.