Question

The values of $a$ and $b$ for which the system $\left[\begin{array}{ccc}3& -2& 1\\ 5& -8& 9\\ 2& 1& a\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}b\\ 3\\ -1\end{array}\right]$
has infinitely many solutions

• A. $a=-3,b=\frac{1}{3}$
• B. $a=3,b=\frac{1}{3}$
• C. $a=-1,b=-\frac{1}{3}$
• D. $a=1,b=\frac{1}{3}$

Answer 1

The correct option is A $a=-3,b=\frac{1}{3}$

Given, the system of equations has infinitely many solutions.
So, $D=D_x=D_y=D_z=0$
$D=0\Rightarrow \left|\begin{array}{ccc}3& -2& 1\\ 5& -8& 9\\ 2& 1& a\end{array}\right|=0$
$\Rightarrow -14a-42=0$
$\Rightarrow a=-3$
Now, taking $D_x=0\Rightarrow \left|\begin{array}{ccc}b& -2& 1\\ 3& -8& 9\\ -1& 1& -3\end{array}\right|=0$
$\Rightarrow 15b-5=0\Rightarrow b=\frac{1}{3}$