For the matrix . Show that A3 – 6A2 + 5A + 11I3 = O.Hence, find A – 1.
A =
A3 = A2.A
A2 =
=
A2.A =
=
=
Now, A3 – 6A2 + 5A + 11I
+ 11
=
=
=
Thus, A3 – 6A2 + 5A + 11I
Now, (AAA) A – 1. – 6(A.A) A – 1 + 5.A A – 1 + 11I.A – 1 = 0
AA(A – 1A) – 6A(A – 1A) + 5(A – 1A) = – 1(A – 1I)
A2 – 6A + 5I = 11 A – 1
= A – 1 =
Now,
A2 – 6A + 5I
=
=
=
=
Hence, A – 1 =