If A -2-11-12-11-12- Verify that A 3-6 A 2-9 A -4 I - O and hence find A-1-

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Byju's Answer

Standard XII

Mathematics

Rational Function

If A =2-11-12...

Question

If $A = [\begin{matrix} 2 & - 1 & 1 \\ - 1 & 2 & - 1 \\ 1 & - 1 & 2 \end{matrix}]$ . Verify that $A^{3} - 6 A^{2} + 9 A - 4 I = O$ and hence find A⁻¹.

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Solution

$A = [2 - 1 1 - 1 2 - 1 1 - 1 2] \Rightarrow |A| = |2 - 1 1 - 1 2 - 1 1 - 1 2| = 2 \times (4 - 1) + 1 (- 2 + 1) + 1 (1 - 2) = 6 - 1 - 1 = 4 Since, |A| \neq 0 Hence, A^{- 1} exists . Now, A^{2} = [2 - 1 1 - 1 2 - 1 1 - 1 2] [2 - 1 1 - 1 2 - 1 1 - 1 2] = [4 + 1 + 1 - 2 - 2 - 1 2 + 1 + 2 - 2 - 2 - 1 1 + 4 + 1 - 1 - 2 - 2 2 + 1 + 2 - 1 - 2 - 2 1 + 1 + 4] = [6 - 5 5 - 5 6 - 5 5 - 5 6] A^{3} = A^{2} . A = [6 - 5 5 - 5 6 - 5 5 - 5 6] [2 - 1 1 - 1 2 - 1 1 - 1 2] = [12 + 5 + 5 - 6 - 10 - 5 6 + 5 + 10 - 10 - 6 - 5 5 + 12 + 5 - 5 - 6 - 10 10 + 5 + 6 - 5 - 10 - 6 5 + 5 + 12] = [22 - 21 21 - 21 22 - 21 21 - 21 22] Now, A^{3} - 6 A^{2} + 9 A - 4 I = [22 - 21 21 - 21 22 - 21 21 - 21 22] - 6 [6 - 5 5 - 5 6 - 5 5 - 5 6] + 9 [2 - 1 1 - 1 2 - 1 1 - 1 2] - 4 [1 0 0 0 1 0 0 0 1] = [22 - 36 + 18 - 4 - 21 + 30 - 9 - 0 21 - 30 + 9 - 0 - 21 + 30 - 9 - 0 22 - 36 + 18 - 4 - 21 + 30 - 9 - 0 21 - 30 + 9 - 0 - 21 + 30 - 9 - 0 22 - 36 + 18 - 4] = [0 0 0 0 0 0 0 0 0] = O [Null matrix] Hence proved . Now, A^{3} - 6 A^{2} + 9 A - 4 I = O \Rightarrow A^{- 1} \times (A^{3} - 6 A^{2} + 9 A - 4 I) = A^{- 1} \times O \Rightarrow A^{2} - 6 A + 9 I - 4 A^{- 1} = O \Rightarrow 4 A^{- 1} = A^{2} - 6 A + 9 I \Rightarrow 4 A^{- 1} = [6 - 5 5 - 5 6 - 5 5 - 5 6] - 6 [2 - 1 1 - 1 2 - 1 1 - 1 2] + 9 [1 0 0 0 1 0 0 0 1] \Rightarrow 4 A^{- 1} = [6 - 12 + 9 - 5 + 6 + 0 5 - 6 + 0 - 5 + 6 + 0 6 - 12 + 9 - 5 + 6 + 0 5 - 6 + 0 - 5 + 6 + 0 6 - 12 + 9] \Rightarrow 4 A^{- 1} = [3 1 - 1 1 3 1 - 1 1 3] \Rightarrow A^{- 1} = \frac{1}{4} [3 1 - 1 1 3 1 - 1 1 3]$

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Similar questions

Q. If

A = ⎡ ⎢ ⎣ \begin{matrix} 2 & - 1 & 1 - 1 & 2 & - 1 1 & - 1 & 2 \end{matrix} ⎤ ⎥ ⎦

, verify that

A^{3} - {6 A}^{2} + 9 A - 4 I = 0

. Hence find

A^{- 1}

If verify that A³ − 6A² + 9A − 4I = O and hence find A⁻¹

Q. If verify that A 3 − 6 A 2 + 9 A − 4 I = O and hence find A −1

If $A = ⎡ ⎢ ⎣ \begin{matrix} 2 & - 1 & 1 - 1 & 2 & - 1 1 & - 1 & 2 \end{matrix} ⎤ ⎥ ⎦$ verify that $A^{3} - 6 A^{2} + 9 A - 4 I = 0$ and hence, find $A^{- 1}$

Q. If

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show that

A^{3} - 6 A^{2} + 5 A + 11 I = 0

. Hence find

A^{- 1}

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If A=2-11-12-11-12. Verify that A3-6A2+9A-4I=O and hence find A−1.

If $A = [\begin{matrix} 2 & - 1 & 1 \\ - 1 & 2 & - 1 \\ 1 & - 1 & 2 \end{matrix}]$ . Verify that $A^{3} - 6 A^{2} + 9 A - 4 I = O$ and hence find A⁻¹.