If for matrix, A =left[begin{array}{cc}1 & -α α & βend{array}right], AA ^{ T }= I

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3 and 4 .Determinants and Matrices

medium

If for the matrix, $A =\left[\begin{array}{cc}1 & -\alpha \\ \alpha & \beta\end{array}\right], AA ^{ T }= I _{2},$ then the value of $\alpha^{4}+\beta^{4}$ is ....... .

$4$

$2$

$3$

$1$

(JEE MAIN-2021)

Solution

$A =\left[\begin{array}{cc}1 & -\alpha \\ \alpha & \beta\end{array}\right] \quad AA ^{ T }= I _{2}$

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

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

$\Rightarrow \alpha^{2}=0 \;and\; \beta^{2}=1$

$\therefore \alpha^{4}+\beta^{4}=1$

Standard 12

Mathematics

Which of the following is incorrect

medium

If $A$ is $3×4$ matrix and $ B$ is a matrix such that $A'B$ and $BA'$ are both defined. Then $B $ is of the type

easy

If $A=\left[\begin{array}{cc}\sqrt{2} & 1 \\ -1 & \sqrt{2}\end{array}\right], B=\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right], C=\mathrm{ABA}^{\mathrm{T}}$ and $\mathrm{X}$ $=\mathrm{A}^{\mathrm{T}} \mathrm{C}^2 \mathrm{~A}$, then $\operatorname{det} \mathrm{X}$ is equal to :

hard

(JEE MAIN-2024)

Which one of the following is correct

normal

If matrix $A = {\left[ {{a_{ij}}} \right]_{3 \times 3}} , B = {\left[ {{b_{ij}}} \right]_{3 \times 3}}$ , where $a_{ij} + a_{ji} = 0$ and $b_{ij} -b_{ji} = 0\, \forall\, i , j$ then $A^4B^3$ is