Question

Calculate the value of the following determinant:

$\left|\begin{array}{cccc}1+a& 1& 1& 1\\ 1& 1+b& 1& 1\\ 1& 1& 1+c& 1\\ 1& 1& 1& 1+d\end{array}\right|$.

Answer 1

$Given\Delta =\left|\begin{array}{cccc}1+a& 1& 1& 1\\ 1& 1+b& 1& 1\\ 1& 1& 1+c& 1\\ 1& 1& 1& 1+d\end{array}\right|$

$R_1=R_1-R_2$

$\Rightarrow \Delta =\left|\begin{array}{cccc}a& -b& 0& 0\\ 1& 1+b& 1& 1\\ 1& 1& 1+c& 1\\ 1& 1& 1& 1+d\end{array}\right|$

Expanding along $R_1$

$\Rightarrow \Delta =a\left|\begin{array}{ccc}1+b& 1& 1\\ 1& 1+c& 1\\ 1& 1& 1+d\end{array}\right|-(-b)\left|\begin{array}{ccc}1& 1& 1\\ 1& 1+c& 1\\ 1& 1& 1+d\end{array}\right|+0-0$

$\Rightarrow \Delta =a\left|\begin{array}{ccc}1+b& 1& 1\\ 1& 1+c& 1\\ 1& 1& 1+d\end{array}\right|+b\left|\begin{array}{ccc}1& 1& 1\\ 1& 1+c& 1\\ 1& 1& 1+d\end{array}\right|$

let ${\Delta }_1=\left|\begin{array}{ccc}1+b& 1& 1\\ 1& 1+c& 1\\ 1& 1& 1+d\end{array}\right|$

$R_1=R_1-R_2{\Delta }_1=\left|\begin{array}{ccc}b& -c& 0\\ 1& 1+c& 1\\ 1& 1& 1+d\end{array}\right|$

${\Delta }_1=b\left\{\right(1+c\left)\right(1+d)-1\}-(-c)\{1+d-1\}+0$

${\Delta }_1=b(1+d+c+cd-1+cd\}$

${\Delta }_1=bd+bc+2bcd$

Let ${\Delta }_2=\left|\begin{array}{ccc}1& 1& 1\\ 1& 1+c& 1\\ 1& 1& 1+d\end{array}\right|$

$R_1=R_1-R_2$

$\Rightarrow {\Delta }_2=\left|\begin{array}{ccc}0& -c& 0\\ 1& 1+c& 1\\ 1& 1& 1+d\end{array}\right|$

$\Rightarrow {\Delta }_2=0-(-c)\{1+d-1\}+0$

$\Rightarrow {\Delta }_2=cd$

Now $\Delta =a{\Delta }_1+b{\Delta }_2$

$\Rightarrow \Delta =a(bd+bc+2bcd)+b\left(cd\right)$

$\Rightarrow \Delta =2abcd+abd+abc+bcd$