91Ó°ÊÓ

If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?

If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?

Find the values of \(x, y\) and \(z\) from the following equations: (i) \(\left[\begin{array}{ll}4 & 3 \\ x & 5\end{array}\right]=\left[\begin{array}{ll}y & z \\ 1 & 5\end{array}\right]\) (ii) \(\left[\begin{array}{cc}x+y & 2 \\ 5+z & x y\end{array}\right]=\left[\begin{array}{cc}6 & 2 \\ 5 & 8\end{array}\right]\) (iii) \(\left[\begin{array}{c}x+y+z \\ x+z \\\ y+z\end{array}\right]=\left[\begin{array}{l}9 \\ 5 \\ 7\end{array}\right]\)

Express the following matrices as the sum of a symmetric and a skew symmetric matrix: (i) \(\left[\begin{array}{rr}3 & 5 \\ 1 & -1\end{array}\right]\) (ii) \(\left[\begin{array}{rrr}6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3\end{array}\right]\) (iii) \(\left[\begin{array}{rrr}3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2\end{array}\right]\) (iv) \(\left[\begin{array}{rr}1 & 5 \\ -1 & 2\end{array}\right]\)