Prove If A is an n x n matrix the homogeneous system Ax 0 h

Prove: If A is an n x n matrix, the homogeneous system Ax = 0 has a nontrivial solution if and only if A is singular.

Solution

Given Ax=0

Let A be a 2x2 matrix

The homogeneous system
ax + by = 0
cx + dy = 0

it can have a trival solution when x=0

If A is an n × n non–singular
matrix, then the homogeneous system
AX = 0 has only the trivial solution X = 0.
Hence if the system AX = 0 has a non–trivial
solution, A is singular.

ex:

EXAMPLE.
A =

1 2 3
1 0 1
3 4 7

is singular. For it can be verified that A has
reduced row–echelon form

1 0 1
0 1 1
0 0 0

and consequently AX = 0 has a non–trivial
solution x = 1, y = 1, z = 1.
REMARK. More generally, if A is
row–equivalent to a matrix containing a zero
row, then A is singular. For then the
homogeneous system AX = 0 has a
non–trivial solution