Question

If the co-factor $C_{21}$ (Cofactor of element of $i^{\text{th}}$ row $\&j^{\text{th}}$ column) of the matrix $\left|\begin{array}{cc}5& 20\\ 0& -1\end{array}\right|$ is $\alpha$ and the value of the determinant is $\beta ,$ then $\beta -\alpha$ is

Answer 1

$C_{ij}=(-1{)}^{i+j}\times M_{ij}$

$A=\left[\begin{array}{cc}5& 20\\ 0& -1\end{array}\right]$
$M_{21}=20$
$C_{21}=(-1{)}^{2+1}\times M_{21}=20\times -1=-20=\alpha$

Now on expanding the determinent
$\left|A\right|=-5=\beta$
Thus $\beta -\alpha =-5+20=15$