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Properties of Determinants

If a matrix A...

Question

If a matrix A is such that $3 A^{3} + 2 A^{2} + 5 A + I = 0$ , then $A^{- 1}$ is equal to

- (3 A^{2} + 2 A + 5)

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3 A^{2} + 2 A + 5

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3 A^{2} - 2 A - 5

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None of these

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Solution

The correct option is A $- (3 A^{2} + 2 A + 5)$
$3 A^{3} + 2 A^{2} + 5 A + I = 0$
$3 A^{3} + 2 A^{2} + 5 A + A A^{- 1} = 0$
$A^{- 1} = - 3 A^{2} - 2 A - 5$

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A^{3} + 2 A^{2} + 5 A + I = 0, then A^{- 1}

- (3 A^{2} + 2 A + 5)

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(a^{2} - 2 a + 3) (a^{2} - 2 a + 5) - 35

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