Question

Let $A$ be a $3\times 3$ matrix with entries from the set of real numbers, If the system of equations $A^{2}X=0$ has a non-trivial solution, then

• A. $AX=0$ has a non-trivial solution.
• B. $AX=0$ does not have a non-trivial solution.
• C. $A$ is a non-singular matrix.
• D. none of these.

Answer 1

Answer: (A)

As $A^{2}X=0$ has a non-trivial solution,
$\det \left(A^2\right)=0\Rightarrow (\text{det}(A){)}^2=0$
$\Rightarrow \det (A)=0$
$\therefore AX=0\text{ has a non-trivial solution. }$