Show that the matrix, A = [(1 0 -1), (-2 -1 2), (3 4 1)] satisfies the equation, A^3 – A^2 – 3A – I3 = O. Hence, find A^–1.

Question

Show that the matrix, \(A=\begin{bmatrix}1&0&-1\\-2&-1&2\\3&4&1\end{bmatrix}\)satisfies the equation, A³ – A² – 3A – I₃ = O. 

Hence, find A^–1.

Answer 1

\(A=\begin{bmatrix}1&0&-1\\-2&-1&2\\3&4&1\end{bmatrix}\)

A³ = A².A

Thus, A³ – A² – 3A – I

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

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

A² – A – 3A – I = 0