If A -31-12- show that A2-5 A-7 I-O use this to find A4

Given: #160;A= #160;31 12Now,A2=AA #8658;A2=31 1231 12 #8658;A2=9 13+2 3 2 1+4 #8658;A2=85 53A2 5A+7I #8658;A2 5A+7I=85 53 531 12+71001 #8658;A2 5 ...

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Standard XII

Mathematics

Properties of Inequalities

If A =31-12, ...

Question

If $A = [\begin{matrix} 3 & 1 \\ - 1 & 2 \end{matrix}]$ , show that A² − 5A + 7I = O use this to find A⁴.

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Solution

$Given : A = [\begin{matrix} 3 & 1 \\ - 1 & 2 \end{matrix}] Now, A^{2} = A A \Rightarrow A^{2} = [\begin{matrix} 3 & 1 \\ - 1 & 2 \end{matrix}] [\begin{matrix} 3 & 1 \\ - 1 & 2 \end{matrix}] \Rightarrow A^{2} = [\begin{matrix} 9 - 1 & 3 + 2 \\ - 3 - 2 & - 1 + 4 \end{matrix}] \Rightarrow A^{2} = [\begin{matrix} 8 & 5 \\ - 5 & 3 \end{matrix}] A^{2} - 5 A + 7 I \Rightarrow A^{2} - 5 A + 7 I = [\begin{matrix} 8 & 5 \\ - 5 & 3 \end{matrix}] - 5 [\begin{matrix} 3 & 1 \\ - 1 & 2 \end{matrix}] + 7 [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] \Rightarrow A^{2} - 5 A + 7 I = [\begin{matrix} 8 & 5 \\ - 5 & 3 \end{matrix}] - [\begin{matrix} 15 & 5 \\ - 5 & 10 \end{matrix}] + [\begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}] \Rightarrow A^{2} - 5 A + 7 I = [\begin{matrix} 8 - 15 + 7 & 5 - 5 + 0 \\ - 5 + 5 + 0 & 3 - 10 + 7 \end{matrix}] \Rightarrow A^{2} - 5 A + 7 I = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}] = 0 Hence proved . Given : A^{2} - 5 A + 7 I = 0 \Rightarrow A^{2} = 5 A - 7 I . . . (1) \Rightarrow A^{3} = A (5 A - 7 I) (Multilpying by A on both sides) \Rightarrow A^{3} = 5 A^{2} - 7 A I \Rightarrow A^{3} = 5 (5 A - 7 I) - 7 A [From eq . (1)] \Rightarrow A^{3} = 25 A - 35 I - 7 A \Rightarrow A^{3} = 18 A - 35 I \Rightarrow A^{4} = (18 A - 35 I) A (Multilpying by A on both sides) \Rightarrow A^{4} = 18 A^{2} - 35 A \Rightarrow A^{4} = 18 (5 A - 7 I) - 35 A [From eq . (1)] \Rightarrow A^{4} = 90 A - 126 I - 35 A \Rightarrow A^{4} = 55 A - 126 I \Rightarrow A^{4} = 55 [\begin{matrix} 3 & 1 \\ - 1 & 2 \end{matrix}] - 126 [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] \Rightarrow A^{4} = [\begin{matrix} 165 & 55 \\ - 55 & 110 \end{matrix}] - [\begin{matrix} 126 & 0 \\ 0 & 126 \end{matrix}] \Rightarrow A^{4} = [\begin{matrix} 165 - 126 & 55 - 0 \\ - 55 - 0 & 110 - 126 \end{matrix}] \Rightarrow A^{4} = [\begin{matrix} 39 & 55 \\ - 55 & - 16 \end{matrix}]$

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Similar questions

Q. (i) If

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, show that A³ − 4A² + A = O.

(ii) If

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, show that A² − 5A + 7I = O use this to find A⁴.

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If A= 31-12, show that A2 − 5A + 7I = O use this to find A4.

If $A = [\begin{matrix} 3 & 1 \\ - 1 & 2 \end{matrix}]$ , show that A² − 5A + 7I = O use this to find A⁴.