[Math] linear equation with that has trivial solution

Question

Let Ax=0 be a homogeneous system of 'n' linear equations in 'n' unknowns that has only the trivial solution. Show that if 'k' is any positive integer, then the system (A^k)x=0 also has only the trivial solution.

Answer 1

If $Ax = 0$, then using the equivalent statements, it is implied that $A$ is invertible since the only solution is the trivial solution. Since $A$ is invertible, $A^{k}$ is invertible as well. It follows that $A^{k}$ has only the trivial solution.