Write minors and cofactors of the elements of following determinants:
(i) \[ \left| \begin{matrix} 2 & -4 \\ 0 & 3 \end{matrix} \right| \]
(ii) \[ \left| \begin{matrix} a & c \\ b & d \end{matrix} \right| \]
(i) \(M_{11}=3,\; M_{12}=0,\; M_{21}=-4,\; M_{22}=2,\; A_{11}=3,\; A_{12}=0,\; A_{21}=4,\; A_{22}=2\)
(ii) \(M_{11}=d,\; M_{12}=b,\; M_{21}=c,\; M_{22}=a\)
\(A_{11}=d,\; A_{12}=-b,\; A_{21}=-c,\; A_{22}=a\)
Write minors and cofactors of the elements of following determinants:
(i) \[ \left| \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} \right| \]
(ii) \[ \left| \begin{matrix} 1 & 0 & 4 \\ 3 & 5 & -1 \\ 0 & 1 & 2 \end{matrix} \right| \]
(i) \(M_{11}=1,\; M_{12}=0,\; M_{13}=0,\; M_{21}=0,\; M_{22}=1,\; M_{23}=0,\; M_{31}=0,\; M_{32}=0,\; M_{33}=1\)
\(A_{11}=1,\; A_{12}=0,\; A_{13}=0,\; A_{21}=0,\; A_{22}=1,\; A_{23}=0,\; A_{31}=0,\; A_{32}=0,\; A_{33}=1\)
(ii) \(M_{11}=11,\; M_{12}=6,\; M_{13}=3,\; M_{21}=-4,\; M_{22}=2,\; M_{23}=1,\; M_{31}=-20,\; M_{32}=-13,\; M_{33}=5\)
\(A_{11}=11,\; A_{12}=-6,\; A_{13}=3,\; A_{21}=4,\; A_{22}=2,\; A_{23}=-1,\; A_{31}=-20,\; A_{32}=13,\; A_{33}=5\)
Using cofactors of elements of second row, evaluate
\[ \Delta = \left| \begin{matrix} 5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3 \end{matrix} \right| \]
7
Using cofactors of elements of third column, evaluate
\[ \Delta = \left| \begin{matrix} 1 & x & yz \\ 1 & y & zx \\ 1 & z & xy \end{matrix} \right| \]
\((x-y)(y-z)(z-x)\)
If
\[ \Delta = \left| \begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix} \right| \]
and \(A_{ij}\) is cofactor of \(a_{ij}\), then value of \(\Delta\) is given by