Question

If $A=\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& a_3\end{array}\right|\ne 0,$ then the system of equations $a_{1}x+b_{1}y+c_{1}z=0,a_{2}x+b_{2}y+c_{2}z=0$ and $a_{3}x+b_{3}y+c_{3}z=0$ has:

• A. only one solution
• B. infinite number of solutions
• C. no solution
• D. more than one but finite number of solutions.

Answer 1

The correct option is A only one solution

We are given the following system of equations.

$a_{1}x+b_{1}y+c_{1}z=0$

$a_{2}x+b_{2}y+c_{2}z=0$

$a_{3}x+b_{3}y+c_{3}z=0$

Now for the given system of equations, we have

$A=\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|$

$X=\left|\begin{array}{c}x\\ y\\ z\end{array}\right|,B=\left|\begin{array}{c}0\\ 0\\ 0\end{array}\right|$

To find AX=B

$B=0\Rightarrow X=0\Rightarrow x=0,y=0,z=0$

So only one solution.