If for the matrix, A= 1 -α α β , AA^T=I_2, then the value of α^4+β^4 is :

Correct answer: (B) 1. A= 1 -α α β AA^T=I_2 ⇒ 1 -α α β 1 α -α β = 1 0 0 1 ⇒ 1+α^2 α-αβ α-αβ 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

Q34 / 90 · JEE Main 2021 (25 Feb Shift 2) · Mathematics

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 :

A
$3$
Correct answer
B
$1$
C
$2$
D
$4$

Solution

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

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

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

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

$\Rightarrow α^{2} = 0 & β^{2} = 1$

$∴ α^{4} = 0 & β^{4} = 1$

$∴ α^{4} + β^{4} = 1$

Hence, the correct answer is option: 'B'.

If for the matrix, A=1-ααβ, AAT=I2, then the value of α4+β4 is :

Solution

Relevant Questions

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 :