# Show that the matrix A = satisfies the equation π³2 + 4π³ β 42 = 0 and hence find A-1.

Given:

To show: Matrix A satisfies the equation x2 + 4x β 42 = 0

If Matrix A satisfies the given equation then

A2 + 4A β 42 = 0

Firstly, we find the A2

Taking LHS of the given equation .i.e.

A2 + 4A β 42

= O

= RHS

β΄ LHS = RHS

Hence matrix A satisfies the given equation x2 + 4x β 42 = 0

Now, we have to find A-1

Finding A-1 using given equation

A2 + 4A β 42 = O

Post multiplying by A-1 both sides, we get

(A2 + 4A β 42)A-1 = OA-1

β A2.A-1 + 4A.A-1 β 42.A-1 = O [OA-1 = O]

β A.(AA-1) + 4I β 42A-1 = O [AA-1 = I]

β A(I) + 4I β 42A-1 = O

β A + 4I β 42A-1 = O

β A + 4I β O = 42A-1

Ans. .

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