\(\left[ \begin{matrix} 1 & a & { a }^{ 2 } \\ 1 & b & { b }^{ 2 } \\ 1 & c & { c }^{ 2 } \end{matrix} \right] =\left| \begin{matrix} 0 & a-b & { a }^{ 2 }-{ b }^{ 2 } \\ 0 & b-c & { b }^{ 2 }-{ c }^{ 2 } \\ 1 & c & { c }^{ 2 } \end{matrix} \right| \begin{matrix} { R }_{ 1 }\rightarrow { R }_{ 1 }-{ R }_{ 2 } \\ { R }_{ 2 }\rightarrow { R }_{ 2 }-{ R }_{ 3 } \end{matrix}\) 
= \(\left( a-b \right) \left( b-c \right) \left| \begin{matrix} 0 & 1 & a+b \\ 0 & 1 & b+c \\ 1 & c & { c }^{ 2 } \end{matrix} \right| \) Take (a - b) and (b - c) from R₁ and R₂ respectively.
= (a - b) (b - c) [(1) (b + c) - (1) (a + b)] = (a - b) (b - c) (c - a)

Since A is a skew-symmetric matrix,
\(\Rightarrow\) |A^T| = -A
\(\Rightarrow\) |A^T| = |-A|
\(\Rightarrow\) |A^T| = (-1)ⁿ |A|  [ \(\therefore\)|kA| = kⁿ|A|]
\(\Rightarrow\) |A| = (-1)ⁿ|A| [\(\therefore\) |A^T| = |A|]
\(\Rightarrow\) |A| = -|A|  [ \(\therefore\)n is odd]
\(\Rightarrow\) |A| = 0
Hence, the determinant of a skew-symmetric matrix of odd order is Zero.

Let  \(\triangle \) = \(\left| \begin{matrix} 41 & 1 & 5 \\ 79 & 7 & 9 \\ 29 & 5 & 3 \end{matrix} \right| \)
Applying C₁ ⟶ C₁ + (-8) C₃ we get
\(\triangle =\left| \begin{matrix} 1 & 1 & 5 \\ 7 & 7 & 9 \\ 5 & 5 & 3 \end{matrix} \right| =0[\therefore { C }_{ 1 }\equiv { C }_{ 2 }]\\ \)
\(\triangle \) = 0

Let A be a symmetric matrix and \(n\in N\)
Then, An A.A.A ..........A upto n-times
\(\Rightarrow\) (Aⁿ)^T [A.A........... A upto n-times]T
\(\Rightarrow\)(Aⁿ)^T A^T.A^T.A^T A^Tupto n-times
\(\Rightarrow\) (Aⁿ)^T = (A^T)ⁿ = Aⁿ [\(\therefore\)  A^T = A]
\(\Rightarrow\) (Aⁿ)^T Aⁿ
Hence Aⁿ is also a symmetric matrix.

Let A be the required column matrix
Then A =\(\begin{matrix} Peons \\ \begin{matrix} Clerks \\ \begin{matrix} Typist \\ Section \ officer \end{matrix} \end{matrix} \end{matrix}\left[ \begin{matrix} 15 \\ \begin{matrix} 6 \\ \begin{matrix} 1 \\ 1 \end{matrix} \end{matrix} \end{matrix} \right] \)
Since there are 30 collegs,
30 A = 30\(\left[ \begin{matrix} 15 \\ \begin{matrix} 6 \\ \begin{matrix} 1 \\ 1 \end{matrix} \end{matrix} \end{matrix} \right] =\left[ \begin{matrix} 450 \\ \begin{matrix} 180 \\ \begin{matrix} 30 \\ 30 \end{matrix} \end{matrix} \end{matrix} \right] \)