91Ó°ÊÓ

Suppose \(A\) is a square matrix. Show (a) \(A+A^{T}\) is symmetric, (b) \(A-A^{T}\) is skew-symmetric, (c) \(A=B+C,\) where \(B\) is symmetric and \(C\) is skew- symmetric.

Suppose \(A\) and \(B\) are symmetric. Show that the following are also symmetric: (a) \(A+B\) (b) \(k A,\) for any scalar \(k\) \((\mathrm{c}) \quad A^{2}\) (d) \(A^{n},\) for \(n>0\) (e) \(f(A),\) for any polynomial \(f(x)\)

Find a \(3 \times 3\) orthogonal matrix \(P\) whose first two rows are multiples of (a) (1,2,3) and (0,-2,3) (b) \(\quad(1,3,1)\) and (1,0,-1)

Find real numbers \(x, y, z\) such that \(A\) is Hermitian, where \(A=\left[\begin{array}{ccc}3 & x+2 i & y i \\ 3-2 i & 0 & 1+z i \\ y i & 1-x i & -1\end{array}\right]\)