If for the matrix, , AAT = I2, then the value of α4 + β4 is :

Engineering

Mathematics

Multiplication of Matrices

Question

If for the matrix, $A = [\begin{matrix} 1 & - α \\ α & β \end{matrix}]$ , AA^T = I₂, then the value of α⁴ + β⁴ is :

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

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

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

⇒ α² = 0 & β² = 1

$∴$ α⁴ + β⁴ = 1