If A = [(3,-5),(-4,2)], then find A^2 – 5A – 14I. Hence, obtain A^3.

Question

If A = \(\begin{bmatrix} 3 & -5 \\[0.3em] -4 & 2 \end{bmatrix}\), then find A² – 5A – 14I. Hence, obtain A³.

Answer 1

By matrix multiplication we can write:

We need to find value of A³ using the above equation:

Now we have,

A² – 5A – 14I = O

⇒ A² = 5A + 14I

Multiplying with A both sides

⇒ A².A = 5A.A + 14IA

⇒ A³ = 5A² + 14A

Using equation 1 we get: