Q. 255.0( 2 Votes )

# Show that the matrix, satisfies the equation, A^{3} – A^{2} – 3A – I_{3} = O. Hence, find A^{–1}.

Answer :

A =

A^{3} = A^{2}.A

A^{2} =

=

A^{2}.A =

=

=

Now, A^{3} – A^{2} – 3A – I

=

=

=

Thus, A^{3} – A^{2} – 3A – I

Now, (AAA) A ^{– 1}. – (A.A) A ^{– 1} – 3.A A ^{– 1} – I.A ^{– 1} = 0

AA(A ^{– 1}A) – A(A ^{– 1}A) – 3(A ^{– 1}A) = – 1(A ^{– 1}I)

A^{2} – A – 3A – I = 0

= A ^{– 1} =

Now,

=

=

=

Hence, A ^{– 1} =

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