Skew Symmetric Matrix

Q.

If A is a symmetric matrix, then matrix M'AM is

1. Diagonal matrix

2. Identity Matrix

3. Symmetric

4. Skew-symmetric

Q. If matrix
$P=\left(\begin{array}{ccc}0& 1& -1\\ 4& -3& 4\\ 3& -3& 4\end{array}\right]=Q+R$, where Q is symmetric matrix and R is skew symmetric matrix. Then find matrix Q and R.

1. $Q=\frac{1}{2}\left[\begin{array}{ccc}1& 5& 2\\ 5& 6& 1\\ 2& 1& 8\end{array}\right],R=\frac{1}{2}\left[\begin{array}{ccc}1& -3& -4\\ 3& 0& 7\\ 4& 5& 0\end{array}\right]$
2. $Q=\frac{1}{2}\left[\begin{array}{ccc}0& 5& 2\\ 5& 6& 4\\ 2& 1& 8\end{array}\right],R=\frac{1}{2}\left[\begin{array}{ccc}0& -3& -4\\ 3& 0& 7\\ 4& 6& 0\end{array}\right]$
3. $Q=\frac{1}{2}\left[\begin{array}{ccc}0& 5& 2\\ 5& -6& 1\\ 2& 1& 8\end{array}\right],R=\frac{1}{2}\left[\begin{array}{ccc}0& 3& 4\\ 3& 0& 7\\ 4& -7& 0\end{array}\right]$
4. $Q=\frac{1}{2}\left[\begin{array}{ccc}0& 5& 2\\ 5& -6& 1\\ 2& 1& 8\end{array}\right],R=\frac{1}{2}\left[\begin{array}{ccc}0& -3& -4\\ 3& 0& 7\\ 4& -7& 0\end{array}\right]$

Q.

If Show that A + A′ is a symmetric matrix

Q. If A is a square matrix of order n, then adj (adj A ) is equal to

1. None of these
2. $|A{|}^{n-1}A$
3. $|A{|}^{n-2}A$
4. $|A{|}^{n}A$

Q. My doubht is that how to find the suitable trignometric identities for the respective sums

Q. If $A,B,C$ are square matrices of same order such that $AB=O$ and $BC=I$ then ${(A+B)}^2{(A+C)}^2=$

1. $O$
2. $2I$
3. $I$
4. $3I$

Q.

If Show that A + A′ is a symmetric matrix

Q.

If A is symmetric (skew-symmetric)matrix , prove that KA is symmetric ( skew-symmetric) matrix.

Q. cot70+4cos70=?