Transpose Conjugate

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

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

3. $\left[\begin{array}{cc}2+2i& 2+4i\\ 2+6i& 2+8i\end{array}\right]$

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

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.

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

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.

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

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

3. $\left[\begin{array}{cc}2+2i& 2+4i\\ 2+6i& 2+8i\end{array}\right]$

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

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.

If $A=3+5x+7x^2,B=-4-7x+x^2,C=4+5x-8x^2$

Find $A+B+C$?