Transpose of a Matrix

Q. If $3A=\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& -2\\ x& 2& y\end{array}\right]$ and $A\cdot A^T=I$, then $x+y$ is equal to

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

Q. If $A=\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$ then AA' =

1. $14$
2. $\left[\begin{array}{c}1\\ 3\\ 3\end{array}\right]$
3. $\left[\begin{array}{ccc}1& 2& 3\\ 2& 4& 6\\ 3& 6& 9\end{array}\right]$
4. $Noneofthese$

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 A is a matrix of order m x n then the order of $A^T$ where, $A^T$ is the transpose of A is

1. m $\times$ n

2. m $\times$ m

3. n $\times$ m

4. n $\times$ n

Q. Let M and N be two 3 $\times$ 3 skew - symmetric matrices such that MN = NM. If $P^T$ denotes the transpose of P, then $M^2{N}^2\left(M^{T}N{)}^{-1}\right(M{N}^{-1}{)}^T$ is equal to

1. $M^2$
2. $-N^2$
3. $-M^2$
4. $MN$

Q.

If $A=\left[\begin{array}{cccc}1& 0& 2& 3\\ 4& 2& 6& 4\end{array}\right]$ then fourth element of second column of $A^T$ =

___

Q.

If A and B are two square matrices which are skew symmetric then$(AB{)}^T$ equals to

1. AB
2. BA
3. -AB
4. -BA

Q.

If A and B are two square matrices which are skew symmetric then$(AB{)}^T$ equals to

1. AB
2. BA
3. -AB
4. -BA

Q. Which of the following are the properties of transpose :-

1. ${\left(A^T\right)}^T=A$
2. ${(A\pm B)}^T=A^T\pm B^T$
3. ${\left(A^T\right)}^2={\left(A^2\right)}^T$
4. ${\left(\mathrm{AB}\right)}^T=B^T{A}^T$