Mathematics - Determinant Question with Solution | TestHub

$D=\left|\begin{array}{ccc}1& 2& -3\\ 2& 6& -11\\ 1& -2& 7\end{array}\right|$

$=20-2\left(25\right)-3(-10)$

$=20-50+30=0$

$D_1=\left|\begin{array}{ccc}a& 2& -3\\ b& 6& -11\\ c& -2& 7\end{array}\right|$

$=20a-2(7b+11c)-3(-2b-6c)$

$=20a-14b-22c+6b+18c$

$=20a-8b-4c$

$=4(5a-2b-c)$

$D_2=\left|\begin{array}{ccc}1& a& -3\\ 2& b& -11\\ 1& c& 7\end{array}\right|$

$=7b+11c-a\left(25\right)-3(2c-b)$

$=7b+11c-25a-6c+3b$

$=-25a+10b+5c$

$=-5(5a-2b-c)$

$D_3=\left|\begin{array}{ccc}1& 2& a\\ 2& 6& b\\ 1& -2& c\end{array}\right|$

$=6c+2b-2(2c-b)-10a$

$=-10a+4b+2c$

$=-2(5a-2b-c)$

for infinite solution

$D=D_1=D_2=D_3=0$

$\Rightarrow 5a=2b+c$