In a certain city there are 30 colleges. Each college has 15 peons, 6 clerks, 1 typist and 1 section of Express the given information as a column matrix. Using sclar multiplication find the total number of each kind in all the colleges.

Show that all positive integral powers of a symmetric are symmetric.

Without expanding evaluate the determinant \(\left| \begin{matrix} 41 & 1 & 5 \\ 79 & 7 & 9 \\ 29 & 5 & 3 \end{matrix} \right| \)

If A is a skew-symmetric matrix of odd order n, then |A| = 0.

Prove that \(\left[ \begin{matrix} 1 & a & { a }^{ 2 } \\ 1 & b & { b }^{ 2 } \\ 1 & c & { c }^{ 2 } \end{matrix} \right] =\left( a-b \right) \left( b-c \right) \left( c-a \right) \)