Singular and Non Singualar Matrices

Q. If $x,y,z$ are three real numbers and $A=\left[\begin{array}{ccc}1& \cos (x-y)& \cos (x-z)\\ \cos (y-x)& 1& \cos (y-z)\\ \cos (z-x)& \cos (z-y)& 1\end{array}\right]$
then
$A$ is

1. symmetric matrix
2. non-singular matrix
3. not invertibe matrix
4. orthogonal matrix

Q.

$IfmatrixA=[a_{ij}{]}_{3\times 3},matrixB=[b_{ij}{]}_{3\times 3}where{a}_{ij}+a_{ji}=0and{b}_{ij}-b_{ji}=0,then|A^4.B^3|is$

1. skew-symmetric matrix

2. singular

3. symmetric

4. zero matrix

Q. If $x,y,z$ are three real numbers and $A=\left[\begin{array}{ccc}1& \cos (x-y)& \cos (x-z)\\ \cos (y-x)& 1& \cos (y-z)\\ \cos (z-x)& \cos (z-y)& 1\end{array}\right]$
then
$A$ is

1. symmetric matrix
2. non-singular matrix
3. not invertibe matrix
4. orthogonal matrix

Q. If
$A=\left[\begin{array}{cccc}{e}^t& e^{-t}\cos t& e^{-t}\sin t& \\ e^t& -e^{-t}\cos t-e^{-t}\sin t& -e^{-t}\sin t+e^{-t}\cos t& \\ e^t& 2e^{-t}\sin t& -2e^{-t}\cos t& \end{array}\right]$,

then $A$ is :

1. invertible only if $t=\pi$.
2. invertible only if $t=\frac{\pi }{2}$.
3. invertible for all $t\in R$.
4. not invertible for any $t\in R$.