Show that the matrix A = [(2,3)(1,2)]satisfies the equation A^2 – 4A + I = O, where I is 2 × 2 identity matrix and O is 2 × 2 zero matrix.

Question

Show that the matrix \(A = \begin{bmatrix} 2 & 3 \\[0.3em] 1&2 \\[0.3em] \end{bmatrix}\)satisfies the equation A² – 4A + I = O, where I is 2 × 2 identity matrix and O is 2 × 2 zero matrix. Using this equation, find A^–1 .

Answer 1

We have,

Hence,

Now,

A² - 4A + I = 0

Therefore,

A.A - 4A = - I

Hence, 

\(A^{-1} = \begin{bmatrix} 2 &-3 \\[0.3em] -1 &2 \\[0.3em] \end{bmatrix}\)