Question

Show that $\left|\begin{array}{ccc}1& bc& a(b+c)\\ 1& ca& b(c+a)\\ 1& ab& c(a+b)\end{array}\right|=0$

Answer 1

The given determinant

$\begin{array}{c}=\left[\begin{array}{ccc}1& bc& a(b+c)\\ 1& ca& b(c+a)\\ 1& ab& c(a+b)\end{array}\right]\\ =\left[\begin{array}{ccc}1& bc& ab+bc+ca\\ 1& ca& ab+bc+ca\\ 1& ab& ab+bc+ca\end{array}\right]\end{array}$ $\left[applying{C}_2\to C_3+C_2\right]$

$=(ab+bc+ca)\left|\begin{array}{ccc}1& bc& 1\\ 1& ca& 1\\ 1& ab& 1\end{array}\right|$ [taking (ab+bc+ca) common from $C_3$]

$=(ab+bc+ca)\times 0=0$ [$C_1$ and $C_3$ are identical]