Question

Write the minor and cofactor of element of the first column of the following matrix and hence evaluate the determinant:

\[A = \begin{bmatrix}1 & a & bc \\ 1 & b & ca \\ 1 & c & ab\end{bmatrix}\]

Answer 1

\[M_{11} = \begin{vmatrix}b & ca \\ c & ab\end{vmatrix} = a b^2 - c^2 a = a\left( b^2 - c^2 \right)\]
\[ M_{21} = \begin{vmatrix}a & bc \\ c & ab\end{vmatrix} = a^2 b - c^2 b = b\left( a^2 - c^2 \right)\]
\[ M_{31 =} \begin{vmatrix}a & bc \\ b & ca\end{vmatrix} = a^2 c - b^2 c = c\left( a^2 - b^2 \right)\]
\[ C_{11} = \left( - 1 \right)^{1 + 1} M_{11} = a\left( b^2 - c^2 \right)\]
\[ C_{21} = \left( - 1 \right)^{2 + 1} M_{21} = - b\left( a^2 - c^2 \right)\]
\[ C_{31} = \left( - 1 \right)^{3 + 1} M_{31} = c\left( a^2 - b^2 \right)\]
\[D = 1 . a\left( b^2 - c^2 \right) - a\left( ab - ca \right) + b . c\left( c - b \right)\]
\[ = a b^2 - a c^2 - a^2 b + a^2 c + c^2 b - b^2 c\]
\[ = a^2 \left( c - b \right) + b^2 \left( a - c \right) + c^2 \left( b - a \right)\]
\[\]