Question

Find the determinants of minors and cofactors of the determinant $\left|\begin{array}{ccc}2& 3& 4\\ 7& 2& -5\\ 8& -1& 3\end{array}\right|$

• A. $\left|\begin{array}{ccc}1& 61& -23\\ 13& -26& -26\\ -23& -38& -17\end{array}\right|$ and $\left|\begin{array}{ccc}1& 61& -23\\ -13& -26& 26\\ -23& 38& -17\end{array}\right|$
• B. $\left|\begin{array}{ccc}1& 61& -23\\ 13& -26& -26\\ -23& -38& -17\end{array}\right|$ and $\left|\begin{array}{ccc}1& -61& 23\\ -13& -26& 26\\ -23& 38& -17\end{array}\right|$
• C. $\left|\begin{array}{ccc}1& 61& -23\\ 13& -26& -26\\ -23& -38& -17\end{array}\right|$ and $\left|\begin{array}{ccc}1& 61& 23\\ -13& -26& 26\\ -23& 38& -17\end{array}\right|$
• D. None of these.

Answer 1

The correct option is D None of these.

Here $M_{11}=\left|\begin{array}{cc}2& -5\\ -1& 3\end{array}\right|$ (Delete 1st row and first column)
$=6-5$
$M_{11}=1$
$\therefore C_{11}=1$ $(\because (-1{)}^{1+1}=1)$
$M_2=\left|\begin{array}{cc}7& -5\\ 8& 3\end{array}\right|$ (Delete 1st row and 2nd column)
$=21-(-40)$
$M_2=61$
$\therefore C_{12}=-61$, $(\because (-1{)}^{1+2}=-1)$
$M_{13}=\left|\begin{array}{cc}7& 2\\ 8& -1\end{array}\right|$ (Delete 1st row and 3rd column)
$=-7-16$
$M_{13}=-23$
$\therefore C_{13}=-23,(\because (-1{)}^{1+3}=1)$
$M_{21}=\left|\begin{array}{cc}3& 4\\ -1& 3\end{array}\right|$ (Delete 2nd row and 1st column)
$=9-(-4)$
$M_{21}=13$
$\therefore C_{21}=-13,(\because (-1{)}^{2+1}=-1)$
$M_{22}=\left|\begin{array}{cc}2& 4\\ 8& 3\end{array}\right|$ (Delete 2nd row and 2nd column)
$=6-32$
$M_{22}=-26$
$\therefore C_{22}=-26,(\because (-1{)}^{2+2}=1)$
$M_{23}=\left|\begin{array}{cc}2& 3\\ 8& -1\end{array}\right|$ (Delete 2nd row and 3rd column)
$=-2-24$
$M_{23}=-26$
$\therefore C_{23}=26,(\because (-1{)}^{2+3}=-1)$
$M_{31}=\left|\begin{array}{cc}3& 4\\ 2& -5\end{array}\right|$ (Delete 3rd row and 1st column)
$=-15-8$
$M_{31}=-23$
$\therefore C_{31}=-23,(\because (-1{)}^{3+1}=1)$
$M_{32}=\left|\begin{array}{cc}2& 4\\ 7& -5\end{array}\right|$ (Delete 3rd row and 2nd column)
$=-10-28$
$M_{32}=-38$
$\therefore C_{32}=38,(\because (-1{)}^{3+2}=-1)$
$M_{33}=\left|\begin{array}{cc}2& 3\\ 7& 2\end{array}\right|$ (Delete 3rd row and 3rd column)
$=4-21$
$M_{33}=-17$
$\therefore C_{33}=-17,(\because (-1{)}^{3+3}=1)$
Hence Determinants of Minors and Cofactors are
$\left|\begin{array}{ccc}1& 61& -23\\ 13& -26& -26\\ -23& -38& -17\end{array}\right|$ and $\left|\begin{array}{ccc}1& -61& -23\\ -13& -26& 26\\ -23& 38& -17\end{array}\right|$