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

for the matrix

 

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

Solution :-

Multiply A with A to get A²

Multiply A² with A again to get A³

Now plug the value in equation A³– 6A² + 5A + 11I = O we get

Hence A³– 6A² + 5A + 11I = O

Multiply equation by A^-1 to get A^-1

We get

A³ A^-1 – 6A² A^-1  + 5A A^-1  + 11I A^-1  = O A^-1

Solve it we get

A²– 6A + 5I + 11 A^-1 = O

11A^-1 =  – A² + 6A – 5I

Plug the value we get