If for the matrix, A = 1 - α α β , A A T = I 2 , then the value of α 4 + β 4 is :

A = 1 - α α β A A T = I 2 ⇒ 1 - α α β 1 α - α β = 1 0 0 1 ⇒ 1 + α 2 α - α β α - α β α 2 + β 2 = 1 0 0 1 ⇒ 1 + α 2 = 1 & α 2 + β 2 = 1 ⇒ α 2 = 0 & β 2 = 1 ∴ α 4 = 0 & β 4 = 1 ∴ α 4 + β 4 = 1

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Mathematics - Matrices Question with Solution | TestHub

MathematicsMatricesProduct of MatricesEasy2 minPYQ_2021

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If for the matrix, $A = [\begin{matrix} 1 & - α \\ α & β \end{matrix}], A A^{T} = I_{2}$ , then the value of $α^{4} + β^{4}$ is :

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Solution:

$A = [\begin{matrix} 1 & - α \\ α & β \end{matrix}] A A^{T} = I_{2}$

$\Rightarrow [\begin{matrix} 1 & - α \\ α & β \end{matrix}] [\begin{matrix} 1 & α \\ - α & β \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$

$\Rightarrow [\begin{matrix} 1 + α^{2} & α - α β \\ α - α β & α^{2} + β^{2} \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$

$\Rightarrow 1 + α^{2} = 1 & α^{2} + β^{2} = 1$

$\Rightarrow α^{2} = 0 & β^{2} = 1$

$∴ α^{4} = 0 & β^{4} = 1$

$∴ α^{4} + β^{4} = 1$

Stream:JEESubject:MathematicsTopic:MatricesSubtopic:Product of Matrices

⏱ 2mℹ️ Source: PYQ_2021

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