Question

The set of equations
$x+y+x=1$
$ax-ay+3z=5$
$5x-3y+az=6$
has infinite soultion, if a=

• A. -4
• B. 3
• C. 4
• D. -3

Answer 1

The correct option is C 4

$\rho \left(A\right)=\rho \left(AB\right)<$ no.of variables
|A|=0
$\left|\begin{array}{cccc}1& 1& 1& \\ a& -a& 3& \\ 5& -3& a& \end{array}\right|=0$
$(-a^2+9)-(a^2-15)+(-3a+5a)=0$
$9-a^2+15-a^2+2a=0$
$2a^2-2a-24=0$
$a^2-a-12=0$
$a^2-4a+3a-12=0$
$(a=4,-3$)
$\rho \left(AB\right)<$(No. of variables)
$\left|\begin{array}{cccc}1& 1& 1& \\ -a& 3& 5& \\ -2& a& 6& \end{array}\right|=0$
$(18-5a)-1(-6a+15)+1(-a^2+9)=0$
$18-5a+15+6a+9-a^2=0$
$a^2-a-12=0$
$a=3,4a=4$
$\left|\begin{array}{ccc}1& 1& 1\\ a& -a& 5\\ 5& -3& 6\end{array}\right|=0$
$(-6a+15)-(6a-25)+(-3a+5a)=0$
$-6a+15-6a+25+21=0$
$10a-=40$
$a=4$
if $a=4|A|=0$
$\left|\begin{array}{ccc}1& 1& 1\\ a& 3& 5\\ 5& a& 6\end{array}\right|=0$
$(18-15)-(6a-25)+(a^2-15)=0$
$a^2-11a+28=0$
$a=7,4$
$a=4$