Question

If for the matrix, $A=\left[\begin{array}{cc}1& -\alpha \\ \alpha & \beta \end{array}\right],A{A}^T=I_2,$ then the value of ${\alpha }^4+{\beta }^4$ is

• A. 4
• B. 2
• C. 3
• D. 1

Answer 1

Answer: (D)

$A=\left[\begin{array}{cc}1& -\alpha \\ \alpha & \beta \end{array}\right],A{A}^T=I_2$
$\Rightarrow \left[\begin{array}{cc}1& -\alpha \\ \alpha & \beta \end{array}\right]\left[\begin{array}{cc}1& \alpha \\ -\alpha & \beta \end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$\Rightarrow \left[\begin{array}{cc}1+{\alpha }^2& \alpha -\alpha \beta \\ \alpha -\alpha \beta & {\alpha }^2+{\beta }^2\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$\Rightarrow {\alpha }^2=0$ & ${\beta }^2=1$
$\therefore {\alpha }^4+{\beta }^4=1$