If a matrix A is such that 4 A^3 + 2 A^2 + 7A + I = O, then A^ - … - Vidyadip

MCQ

If a matrix $A$ is such that $4{A^3} + 2{A^2} + 7A + I = O$, then ${A^{ - 1}}$ equals

A
$(4{A^2} + 2A + 7I)$
✓
$ - (4{A^2} + 2A + 7I)$
C
$ - (4{A^2} - 2A + 7I)$
D
$(4{A^2} + 2A - 7I)$

✓

Answer

Correct option: B.

$ - (4{A^2} + 2A + 7I)$

b
(b) Given $4{A^3} + 2{A^2} + 7A + I = O$
Pre-multiply with ${A^{ - 1}}$
==> ${A^{ - 1}}[4{A^3} + 2{A^2} + 7A + I] = O$
==> $4I{A^2} + 2IA + 7I + {A^{ - 1}}I = O.{A^{ - 1}}$
==> $I(4{A^2} + 2A + 7) + {A^{ - 1}}I = O$
==> ${A^{ - 1}} = - (4{A^2} + 2A + 7I)$.

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