Question

The linear system of equation $\begin{array}{c}8x-3y-5z=0\\ 5x-8y+3z=0\\ 3x+5y-8z=0\end{array}\}$ has

• A. Only 'zero solution'
• B. Only finite number of non-zero solutions
• C. No non-zero solution
• D. Infinitely many non-zero solutions

Answer 1

The correct option is D Infinitely many non-zero solutions

Given system of equations are

$\begin{array}{c}8x-3y-5z=0\\ 5x-8y+3z=0\\ 3x+5y-8z=0\end{array}$

To find the solution, let's find their coefficient determinant

$D=\left|\begin{array}{ccc}8& -3& -5\\ 5& -8& 3\\ 3& 5& -8\end{array}\right|=8(64-15)+3(-40-9)-5(25+24)=0$

$D_1=\left|\begin{array}{ccc}-3& -5& 0\\ -8& 3& 0\\ 5& -8& 0\end{array}\right|=-3(0-0)+5(0-0)+0=0$
Similarly,

$D_2=\left|\begin{array}{ccc}8& -5& 0\\ 5& 3& 0\\ 3& -8& 0\end{array}\right|=0$

and $D_3=0$

Thus, the linear system of equation has infinite solutions.