For the matrix A = [(1 1 1), (1 2 -3), (2 -1 3)]. Show that A^3 – 6A^2 + 5A + 11I3 = O. Hence, find A^–1.

Question

For the matrix \(A=\begin{bmatrix}1&1&1\\1&2&-3\\2&-1&3\end{bmatrix}\). Show that A³ – 6A² + 5A + 11I₃ = O. 

Hence, find A^–1.

Answer 1

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

A³ = A².A

A² =\(\begin{bmatrix}1&1&1\\1&2&-3\\2&-1&3\end{bmatrix}\)\(\begin{bmatrix}1&1&1\\1&2&-3\\2&-1&3\end{bmatrix}\)\(=\begin{bmatrix}1+1+2&1+2-1&1-3+3\\1+2-6&1+4+3&1-6-9\\2-1+6&2-2-3&2+3+9\end{bmatrix}\)\(=\begin{bmatrix}4&2&1\\-3&8&-14\\7&-3&14\end{bmatrix}\)

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