Solving homogeneous system of linear equation

Question

Let A be a nxn matrix, x be a nx1 matrix and 0 be a nx1 zero matrix.
Ax=0

a)If A is invertible, solve the homogeneous system of linear equations

b)Is Ax = 0 always consistent? Explain

Can anyone start me off how to go about to solving this? I am not too sure about a)

For b), I wrote Yes as a homogeneous equation is always consistent.There is always a solution of x=0, hence it will always be consistent.Am I right?

Answer 1

a) If A is invertible, it means that $Ax = 0$, for $x$ only has $0$ for the answer.

You can use gaussian elimination or inverse matrix to solve it.

$Ax = 0$

$A^{-1}Ax = A^{-1}×0$

$x = 0$

b) Of course $Ax = 0$ always has the answer, and that is $0$.

Reference

$A$ is invertible matrix iff $Ax=0$ has the trivial solution only.