Example of solving a 3-by-3 homogeneous system of linear equations by row-reducing the augmented matrix

Example: Solving a 3-by-3 Homogeneous System by Row Reduction

Solve the homogeneous system:

 x + 2y + 3z = 0
2x + 5y + 3z = 0
 x + 8z = 0

Step 1: Write augmented matrix (right column is always 0 for homogeneous systems)

[ 1  2  3 | 0]
[ 2  5  3 | 0]
[ 1  0  8 | 0]

Step 2: R₂ → R₂ - 2R₁, R₃ → R₃ - R₁

[ 1  2  3 | 0]
[ 0  1 -3 | 0]
[ 0 -2  5 | 0]

Step 3: R₃ → R₃ + 2R₂

[ 1  2  3 | 0]
[ 0  1 -3 | 0]
[ 0  0 -1 | 0]

Step 4: R₃ → -R₃

[ 1  2  3 | 0]
[ 0  1 -3 | 0]
[ 0  0  1 | 0]

Step 5: Back-substitute: z = 0, y = 3(0) = 0, x = -2(0) - 3(0) = 0

Solution: The only solution is the trivial solution x = 0, y = 0, z = 0.

This homogeneous system has only the trivial solution because the coefficient matrix has full rank (3 pivots). When a homogeneous system has free variables, nontrivial solutions exist.