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If a matrix A is such that 3A3 + 2A2 + 5A + I = 0 then what is A-1 equal to?

If a matrix A is such that 3A3 + 2A2 + 5A + I = 0 then what is A-1 equal to? Correct Answer -(3A<sup>2</sup> + 2A + 5I)

Concept:

Let A be the square matrix.

A-1A = I and A-1I = A-1

 

Calculations:

Given, a matrix A is such that 3A3 + 2A2 + 5A + I = 0

Consider, 3A3 + 2A2 + 5A + I = 0

Pre - Multiply the above polynomial by A-1.

⇒A-1(3A3 + 2A2 + 5A + I) = A-1(0)

⇒3A-1A3 + 2A-1A2 + 5A-1A + A-1I = 0

⇒3A-1AA2 + 2A-1AA + 5A-1A + A-1I = 0

We know that A-1A = I and A-1I = A-1

⇒3IA2 + 2IA + 5I + A-1 = 0

⇒3A2 + 2A + 5I + A-1 = 0

⇒3A2 + 2A + 5I + A-1 = 0

⇒A-1 =  - (3A2 + 2A + 5I)

 

Hint

To find the value of 3A3 + 2A2 + 5A + I = 0, Pre - Multiply the above polynomial by A-1. 

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