If A=left[begin{array}{cc}sin α & cos α -cos α & sin αend{array}right], then verify that

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3 and 4 .Determinants and Matrices

medium

If $A=\left[\begin{array}{cc}\sin \alpha & \cos \alpha \\ -\cos \alpha & \sin \alpha\end{array}\right],$ then verify that $A^{\prime} A=I$.

Option A

Option B

Option C

Option D

Solution

$ A=\left[\begin{array}{cc}\sin \alpha & \cos \alpha \\ -\cos \alpha & \sin \alpha\end{array}\right]$

$\therefore $ $A^{\prime}=\left[\begin{array}{ll}\sin \alpha & -\cos \alpha \\ \cos \alpha & \sin \alpha\end{array}\right]$

$A^{\prime} A=\left[\begin{array}{ll}\sin \alpha & -\cos \alpha \\ \cos \alpha & \sin \alpha\end{array}\right]\left[\begin{array}{cc}\sin \alpha & \cos \alpha \\ -\cos \alpha & \sin \alpha\end{array}\right]$

$\left[\begin{array}{cc}\sin \alpha & -\cos \alpha \\ \cos \alpha & \sin \alpha\end{array}\right]\left[\begin{array}{cc}\sin \alpha & \cos \alpha \\ -\cos \alpha & \sin \alpha\end{array}\right]$

$=\left[\begin{array}{cc}(\sin \alpha)(\sin \alpha)+(-\cos \alpha)(-\cos \alpha) & (\sin \alpha)(\cos \alpha)+(-\cos \alpha)(\sin \alpha) \\ (\cos \alpha)(\sin \alpha)+(\sin \alpha)(-\cos \alpha) & (\cos \alpha)(\cos \alpha)+(\sin \alpha)(\sin \alpha)\end{array}\right]$

$=\left[\begin{array}{cc}\sin ^{2} \alpha+\cos ^{2} \alpha & \sin \alpha \cos \alpha-\sin \alpha \cos \alpha \\ \sin \alpha \cos \alpha-\sin \alpha \cos \alpha & \cos ^{2} \alpha+\sin ^{2} \alpha\end{array}\right]$

$=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]=I$

Hence, we have verified that $A ^{\prime}A=1$.

Standard 12

Mathematics

If the matrix $\mathrm{A}$ is both symmetric and skew symmetric, then

easy

If $A$ is skew symmetric matrix of order $3$ and $X$ be another matrix of same order, then $|XA + AX^T|$ is (where $|P|$ denotes determinant of matrix $P$ )

normal

Let $A , B , C$ be $3 \times 3$ matrices such that $A$ is symmetric and $B$ and $C$ are skew-symmetric.Consider the statements

$(S1): A ^{13} B ^{26}- B ^{26} A ^{13}$ is symmetric

$(S2):A ^{26} C ^{13}- C ^{13} A ^{26}$ is symmetric

Then,

hard

(JEE MAIN-2023)

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 ……. .

medium

(JEE MAIN-2021)

Consider the matrices $A =$ $\left[ {\begin{array}{{20}{c}}4&6&{ – 1}\\3&0&2\\1&{ – 2}&5\end{array}} \right]$ , $B =$ $\left[ {\begin{array}{{20}{c}}2&4\\0&1\\{ – 1}&2\end{array}} \right]$ , $C =$ $\left[{\begin{array}{*{20}{c}}3\\1\\2\end{array}} \right]$ . Out of the given matrix products
$(i)$ $(AB)^TC$ $(ii)$ $C^TC(AB)^T$ $(iii)$ $C^TAB$ and $(iv)$ $A^TABB^TC$