If A is a 3 - 3 non singular matrix and A 3-3 A 2-2 A -0 then find A -1-A- 3-2 A - I B- I-2 A -4 I C- I-3 A -2 I D- 1-2 A -3 I

The correct option is D 1/2(A+3I)A^3+3A^2 2A=0 Multiply both sides by A^ 1 A^ 1AA^2+3A^ 1AA 2A^ 1A =0 ⇒ A^2+3A 2I=0 Again multiply both sides by A^ 1 A^ 1AA+3A ...

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Byju's Answer

Standard XII

Mathematics

Inverse of a Matrix

If A is a 3 ×...

Question

If $A$ is a $3 \times 3$ non singular matrix and $A^{3} + 3 A^{2} - 2 A = 0$ then find $A^{- 1}$ .

\frac{3}{2} (A + I)

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\frac{1}{2} (A + 4 I)

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\frac{1}{3} (A + 2 I)

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\frac{1}{2} (A + 3 I)

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Solution

The correct option is D $\frac{1}{2} (A + 3 I)$
$A^{3} + 3 A^{2} - 2 A = 0$
Multiply both sides by $A^{- 1}$
$A^{- 1} A A^{2} + 3 A^{- 1} A A - 2 A^{- 1} A = 0 \Rightarrow A^{2} + 3 A - 2 I = 0$
Again multiply both sides by $A^{- 1}$
$A^{- 1} A A + 3 A^{- 1} A - 2 A^{- 1} I = 0 \Rightarrow A + 3 I - 2 A^{- 1} = 0 \Rightarrow A^{- 1} = \frac{1}{2} (A + 3 I)$

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If A is a 3×3 non singular matrix and A3+3A2−2A=0 then find A−1.

The correct option is D 12(A+3I)A3+3A2−2A=0 Multiply both sides by A−1 A−1AA2+3A−1AA−2A−1A=0⇒A2+3A−2I=0 Again multiply both sides by A−1 A−1AA+3A−1A−2A−1I=0⇒A+3I−2A−1=0⇒A−1=12(A+3I)

If $A$ is a $3 \times 3$ non singular matrix and $A^{3} + 3 A^{2} - 2 A = 0$ then find $A^{- 1}$ .