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