If A = [(3,2)(2,1)] verify that A^2 – 4A – I = O, and hence find A^-1.

Question

If A = \(\begin{bmatrix} 3& 2 \\[0.3em] 2 &1 \\[0.3em] \end{bmatrix} \), verify that A² – 4A – I = O, and hence find A^-1.

A = [(3,2)(2,1)]

Answer 1

Given:

To verify: A² – 4 A-1 = 0

Firstly, we find the A²

Taking LHS of the given equation .i.e.

A² – 4A - 1

= 0

= RHS

∴ LHS = RHS

Hence verified 

Now, we have to find A^-1 

Finding A^-1 using given equation 

A² – 4A – I = O 

Post multiplying by A^-1 both sides, we get 

(A² – 4A – I)A^-1 = OA^-1 

⇒ A².A^-1 – 4A.A^-1 – I.A^-1 = O [OA -1 = O] 

⇒ A.(AA^-1) – 4I – A^-1 = O [AA^-1 = I] 

⇒ A(I) – 4I – A^-1 = O 

⇒ A – 4I – A^-1 = O 

⇒ A – 4I – O = A^-1 

⇒ A – 4I = A^-1

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