Applied Course | Mock GATE | Test 1 | Question: 35

Question

If $S=a+b+c$, then the value of $\Delta=\begin{vmatrix} s+c & a & b \\ c & s+a & b \\ c & a & s+b \end{vmatrix}$

• A. $2s^2$
• B. $2s^3$
• C. $s^3$
• D. $3s^3$

Answer 1

Given matrix is
$\begin{vmatrix} s+c & a & b \\ c & s+a & b \\ c & a & s+b \end{vmatrix}$
$r1=r1-r2$
$\begin{vmatrix} s+c-c & a-(s+a) & b-b \\ c & s+a & b \\ c & a & s+b \end{vmatrix}$
$\Rightarrow \begin{vmatrix} s & -s & 0 \\ c & s+a & b \\ c & a & s+b \end{vmatrix}$
$\Rightarrow c2=c1+c2$
$\begin{vmatrix} s & 0 & 0 \\ 0 & s & -s \\ c & a+c & s+b \end{vmatrix}$
$\Rightarrow s(s(s+b)) - (-s(a+c))$
$s(ss+sb+sa+sc)$
$s(ss+ss)$
$s(2s^2)$
$2s^3$