Let Ax rightarrow 0 be an n times n homogeneous system with

Let Ax rightarrow = 0 be an n times n homogeneous system with only the trivial solution. Show that A^3x rightarrow = 0 has only the trivial solution.

Solution

Solution:

Ax=0 has trivial solution if and only if A is invertible.

That is if and only if A^-1 exist.

Given A³x=0  

implies, (A^-1)³.A³x= (A-1)³.0

implies (A^-1.A)³x= 0

implies Ix=0 ,I is the identity matrix.

implies x=0

Thus, A³x=0 has only trivial solution.