Question

Determine 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 no solution

• A. $a=-3andb\ne 1/3$
• B. $a=3andb\ne 1/3$
• C. $a=3andb=1/3$
• D. $a\ne 3andb=1/3$

Answer 1

The correct option is A $a=-3andb\ne 1/3$

Given system of equations are $\left[\begin{array}{ccc}3& -2& 1\\ 5& -8& 9\\ 2& 1& a\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}b\\ 3\\ -1\end{array}\right]$
system of equations has no solution if rank of coefficient matrix and augumented matrix are diffferent.
$\Rightarrow \left|\begin{array}{ccc}3& -2& 1\\ 5& -8& 9\\ 2& 1& a\end{array}\right|=0$
$\Rightarrow 3(-8a-9)+2(5a-18)+21=-14a-42=0$
$\therefore a=-3$
$\Rightarrow \left|\begin{array}{ccc}3& -2& b\\ 5& -8& 3\\ 2& 1& -1\end{array}\right|\ne 0$
$\Rightarrow -7+21b\ne 0$
$\therefore b\ne 1/3$
Hence, given system has no solution if $a=-3$ and $b\ne 1/3$
Hence, option A.