Transpose Conjugate

Q. If the product of matrices $A=\left[\begin{array}{cc}{\cos }^2\theta & \cos \theta \sin \theta \\ \cos \theta \sin \theta & {\sin }^2\theta \end{array}\right],B=\left[\begin{array}{cc}{\cos }^2\varphi & \cos \varphi \sin \varphi \\ \cos \varphi \sin \varphi & {\sin }^2\varphi \end{array}\right]$ is a null matrix, then $\theta -\varphi$ is equal to

1. $2n\pi ,n\in Z$
2. $\frac{n\pi }{2},n\in Z$
3. $(2n+1)\frac{\pi }{2},n\in Z$
4. $n\pi ,n\in Z$

Q. 20. For what value of x the matrix A is singular? A=[x-1 1 1] [1 x-1 1] [1 1 x-1]

Q. If $A$ is a non-singular skew-symmetric matrix and $B$ is a square matrix such that $B={\left(\right(A^T{B}^T\left)A^{-1}\right)}^T,$ then $(A+B{)}^2$ is equal to

1. $O$
2. $A+B$
3. $A^2+B^2$
4. $A^2+B^2+2AB$

Q. How to solve a cubic equation with eg

Q. Prove that the inverse of a given square matrix, if it exists, is unique

Q. Let P, Q, R, S be (not necessarily square) real matrices such that
$P^T=\mathrm{QRS};Q^T=\mathrm{RSP};R^T=\mathrm{SPQ}\mathrm{and}{S}^T=\mathrm{PQR}.$
If the matrix A = PQRS, then $A^3=?$

1. A
2. $A^2$
3. $A^6$
4. $A^9$

Q.

What is the transpose conjugate for the matrix. $\left[\begin{array}{cc}2+3i& -i\\ 5& 5-i\end{array}\right]$

1. 

2. 

3. 

4. 

Q. If the determinant $\left|\begin{array}{ccc}1+{\sin }^2\theta & {\sin }^2\theta & {\sin }^2\theta \\ {\cos }^2\theta & 1+{\cos }^2\theta & {\cos }^2\theta \\ 4\sin 4\theta & 4\sin 4\theta & 1+4\sin 4\theta \end{array}\right|=0$,
then $\sin 4\theta$ equals to
$(\text{where}\theta \in [0,2\pi \left]\right)$

1. $\frac{1}{2}$
2. $1$
3. $-\frac{1}{2}$
4. $-1$

Q.

Given $2A^{\ast }+B^{\ast }=\left[\begin{array}{cc}-2i& -4i\\ -6i& -8i\end{array}\right]$ and all the elements of B are 2. Then matrix A is given by. ($\ast$ represents transpose conjugate operator)

1. 

2. 

3. 

4. 

Q.

How do you find the diagonal of a rectangle whose sides are $20$ and $48$?

Q. Let $A$ and $B$ be any two $3\times 3$ matrices. If $A$ is symmetric and $B$ is skew-symmetric, then the matrix $AB-BA$ is always :

1. Skew symmetric
2. Symmetric
3. Neither symmetric nor skew symmetric
4. Lower triangular matrix

Q. If $A$ and $B$ are symmetric matrices of the same order and $X=AB+BA$ and $Y=AB-BA,$ then $X{Y}^T$ is equal to

1. $XY$
2. $YX$
3. $-XY$
4. None of these

Q. If $\left[\begin{array}{cc}\sin \theta & -\cos \theta \\ \cos \theta & \sin \theta \end{array}\right]\left[\begin{array}{cc}\sin \theta & \cos \theta \\ -\cos \theta & \sin \theta \end{array}\right]=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],$ then the value of $a+b+c+d$ is

1. $0$
2. $\sin \theta +\cos \theta +1$
3. $2$
4. $1$

Q.

If A and B are 2 matrices given by general elements $a_{ij}$ and $b_{ij}$ respectively. And $b_{ij}$ = Im($a_{ij}$). Then which the following are correct if A* + B = 0. ( Im(x) means Imaginary part of the number x)

1. A is a symmetric matrix

2. A is a non symmetric matrix

3. Sum of real parts of elements of A is zero

4. Sum of imaginary parts of elements of A is zero

Q.

What is the transpose conjugate for the matrix. $\left[\begin{array}{cc}2+3i& -i\\ 5& 5-i\end{array}\right]$

1. 

2. 

3. 

4. 

Q. Prove the below property of idemp. matrix If AB=A nad BA=B then A^n+B^n=A+B

Q. $IfA=\left[\begin{array}{cc}cos\theta & sin\theta \\ sin\theta & -cos\theta \end{array}\right],B=\left[\begin{array}{cc}1& 0\\ -1& 1\end{array}\right],C=AB{A}^T,then{A}^T{C}^{n}Aequalsto\left(nϵZ^{+}\right)$

1. $\left[\begin{array}{cc}-n& 1\\ 1& 0\end{array}\right]$
2. $\left[\begin{array}{cc}1& -n\\ 0& 1\end{array}\right]$
3. $\left[\begin{array}{cc}0& 1\\ 1& -n\end{array}\right]$
4. $\left[\begin{array}{cc}1& 0\\ -n& 1\end{array}\right]$

Q. If A and B are symmetric matrices of same order and X= AB + BA and Y = AB – BA, then $(XY{)}^T$ is equal to

1. XY
2. YX
3. – YX
4. None of these

Q.

Differentiate the function with respect to x,

x inverse x cos x+x square +1/x square -1

Q.

What is the transpose conjugate for the matrix. $\left[\begin{array}{cc}2+3i& -i\\ 5& 5-i\end{array}\right]$

1. $\left[\begin{array}{cc}2-3i& -5\\ -i& 5+i\end{array}\right]$

2. $\left[\begin{array}{cc}-2-3i& -5\\ i& -5+i\end{array}\right]$

3. $\left[\begin{array}{cc}5-i& -i\\ 5& 2+3i\end{array}\right]$

4. $\left[\begin{array}{cc}2-3i& 5\\ i& 5+i\end{array}\right]$

Q. If A and B are symmetric matrices of same order and X= AB + BA and Y = AB – BA, then $(XY{)}^T$ is equal to

1. XY
2. YX
3. – YX
4. None of these

Q. If the adjoint of a $3\times 3$ matrix P is $\left[\begin{array}{ccc}1& 4& 4\\ 2& 1& 7\\ 1& 1& 3\end{array}\right]$, then the possible value(s) of the determinant of P is (are)

1. $-1$
2. $-2$
3. $2$
4. $1$

Q. $IfA=\left[\begin{array}{cc}cos\theta & sin\theta \\ sin\theta & -cos\theta \end{array}\right],B=\left[\begin{array}{cc}1& 0\\ -1& 1\end{array}\right],C=AB{A}^T,then{A}^T{C}^{n}Aequalsto\left(nϵZ^{+}\right)$

1. $\left[\begin{array}{cc}-n& 1\\ 1& 0\end{array}\right]$
2. $\left[\begin{array}{cc}1& -n\\ 0& 1\end{array}\right]$
3. $\left[\begin{array}{cc}0& 1\\ 1& -n\end{array}\right]$
4. $\left[\begin{array}{cc}1& 0\\ -n& 1\end{array}\right]$

Q.

What is the transpose conjugate for the matrix. $\left[\begin{array}{cc}2+3i& -i\\ 5& 5-i\end{array}\right]$

1. $\left[\begin{array}{cc}2-3i& -5\\ -i& 5+i\end{array}\right]$

2. $\left[\begin{array}{cc}-2-3i& -5\\ i& -5+i\end{array}\right]$

3. $\left[\begin{array}{cc}5-i& -i\\ 5& 2+3i\end{array}\right]$

4. $\left[\begin{array}{cc}2-3i& 5\\ i& 5+i\end{array}\right]$

Q. If $A=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$, then $A{A}^T$ equals

1. $\left[\begin{array}{cc}{\cos }^2\theta & {\sin }^2\theta \\ {\sin }^2\theta & {\cos }^2\theta \end{array}\right]$
2. $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
3. $\left[\begin{array}{cc}\cos 2\theta & -\sin 2\theta \\ \sin 2\theta & \cos 2\theta \end{array}\right]$
4. $\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

Q. 81.a(a+b+c)-bc

Q. If $A=\left[\begin{array}{cc}\cos 2\theta & -\sin 2\theta \\ \sin 2\theta & \cos 2\theta \end{array}\right]$ and $A+A^T=I,$ where $I$ is $2\times 2$ unit matrix and $A^T$ is the transpose of $A,$ then the value of $\theta$ is equal to

1. $\frac{3\pi }{2}$
2. $\frac{\pi }{2}$
3. $\frac{\pi }{3}$
4. $\frac{\pi }{6}$