Question

Consider the system of equations
$a_{1}x+b_{1}y+c_{1}z=d_1$
$a_{2}x+b_{2}y+c_{2}z=d_2$
$a_{3}x+b_{3}y+c_{3}z=d_3$
This system of equations can be written as $AX=B$, where $A=\left[\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right].X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$ and $B=\left[\begin{array}{c}{d}_1\\ d_2\\ d_3\end{array}\right]$.
If $A$ is singular and $(\mathrm{adj}A)B=0$, then the system $AX=B$ has

• A. infinitely maryy solution or no solution
• B. no solution only
• C. unique solution
• D. None of the above

Answer 1

Answer: (A)

If $A$ is a singular matrix, then $|A|=0$ and if $(\text{adj}A)B=O$, then the system may be either consistent or inconsistent according as the system have either infinitely many solutions or no solution.