Question

The value of $\Delta=\begin{vmatrix}5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3\end{vmatrix}$ using cofactors of elements of second row is

• A. $-14$
• B. 14
• C. $-7$
• D. 7

Answer 1

Answer: (D)

Given, $\Delta=\begin{vmatrix}5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3\end{vmatrix}$
Cofactors of the elements of second row are
$A_{21}=(-1)^{2+1}\begin{vmatrix}3 & 8 \\2 & 3 \end{vmatrix}=-(9-16)=7$
$\left[\because A_{i j}=(-1)^{j+i} M_{i j}\right]$
$A_{22}=(-1)^{2+2}\begin{vmatrix}5 & 8 \\ 1 & 3\end{vmatrix}=15-8=7$
and $\quad A_{23}=(-1)^{2+3}\begin{vmatrix}5 & 3 \\ 1 & 2\end{vmatrix}=-(10-3)=-7$
Now, expansion of $\Delta$ using cofactors of elements of second row is given by
$ \Delta =a_{21} A_{21}+a_{22} A_{22}+a_{23} A_{23}$
$ =2 \times 7+0 \times 7+1(-7) $
$ =14-7=7$