Symmetric Matrix

Q. The number of symmetric matrices of order $3\times 3$ all of whose elements are either 0 or 1 is

1. 256
2. 128
3. 64
4. 32

Q. Out of the following which is a skew- symmetric matrix

1. $\left[\begin{array}{ccc}0& 4& 5\\ -4& 0& -6\\ -5& 6& 0\end{array}\right]$
2. $\left[\begin{array}{ccc}1& 4& 5\\ -4& 1& -6\\ -5& 6& 1\end{array}\right]$
3. $\left[\begin{array}{ccc}1& 4& 5\\ -4& 2& -6\\ -5& 6& 3\end{array}\right]$
4. $\left[\begin{array}{ccc}i+1& 4& 5\\ -4& i& -6\\ -5& 6& i\end{array}\right]$

Q.

Are Eigenvectors Orthogonal?

Q.

What is a symmetric and skew-symmetric matrix?

Q. $\mathrm{Any}\mathrm{matrix}\mathrm{can}\mathrm{be}\mathrm{represented}\mathrm{as}a\mathrm{sum}\mathrm{of}a\mathrm{symmetric}\mathrm{matrix}\mathrm{and}a\mathrm{skew}\mathrm{symmetric}\mathrm{matrix}$

1. False
2. True

Q. Let A be a non singular, symmetric matrix of order three such that $A=adj(A+A^T)$, then

1. $A^{-1}=\frac{A}{16}$
2. $A^{-1}=\frac{A}{64}$
3. $adj\left(A^{-1}\right)=64A$
4. $adj\left(A^{-1}\right)=\frac{A}{1024}$

Q.

If A is a square matrix $\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$ then $A+A^T$ is a symmetric matrix.

1. True

2. False

Q. Let $A$ and $B$ be $3\times 3$ real matrices such that $A$ is symmetric matrix and $B$ is skew-symmetric matrix. Then the system of linear equations $(A^2{B}^2-B^2{A}^2)X=O,$ where $X$ is a $3\times 1$ column matrix of unknown variables and $O$ is a $3\times 1$ null matrix, has

1. a unique solution
2. exactly two solutions
3. infinitely many solutions
4. no solution