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

Q14 of 127 Page 122

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

We have A² = A.A =

Since A² + aA + bI =

So A² + aA + bI =

Hence 10+3a+b = 0 …(i)

5+a = 0 …(ii)

5+2a+b = 0 …(iii)

From (ii) a = -5

Putting a in (iii) we get b = 5

So a = -5 and b = 5 satisfy the equation.

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

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

For the matrix

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

If Verify that A³ – 6A² + 9A – 4I = O and hence find A^-1.