Question

15. 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 : (1) 4 (2) 2 (3) 3 (4) 1

Answer 1

Correct option is (4) 1

\( \mathrm{A}=\left[\begin{array}{cc} 1 & -\alpha \\ \alpha & \beta \end{array}\right] \)  \(\mathrm{AA}^{\mathrm{T}}=\mathrm{I}_2 \)

\( \Rightarrow\left[\begin{array}{cc} 1 & -\alpha \\ \alpha & \beta \end{array}\right]\left[\begin{array}{cc} 1 & \alpha \\ -\alpha & \beta \end{array}\right]=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \)

\(\Rightarrow\left[\begin{array}{cc} 1+\alpha^2 & \alpha-\alpha \beta \\ \alpha-\alpha \beta & \alpha^2+\beta^2 \end{array}\right]=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \)

\( \Rightarrow \alpha^2=0 \,\& \,\beta^2=1\)

\( \therefore \alpha^4+\beta^4=1\)