2.6 Homogeneous Systems of Linear Equations

A homogeneous system of linear equations has the form $AX=0$, where all constant terms on the right-hand side are zero.

General Form

A system of three homogeneous equations in three unknowns:

$a_{1}x+b_{1}y+c_{1}z=0$ $a_{2}x+b_{2}y+c_{2}z=0$ $a_{3}x+b_{3}y+c_{3}z=0$

In matrix form: $AX=0$, where $A$ is the $3\times 3$ coefficient matrix and $X=[x,y,z{]}^T$.

Trivial and Non-Trivial Solutions

Solving by Gaussian Elimination (SLO M-11-A-20)

Gaussian elimination reduces the augmented matrix $[A|0]$ to row echelon form using elementary row operations.

Steps:

1. Write the augmented matrix $[A|0]$.
2. Apply row operations ($R_i\leftrightarrow R_j$, $k{R}_i$, $R_i+k{R}_j$) to get an upper triangular form.
3. Determine the rank of $A$:
   • If $\text{rank}(A)=n$ → only trivial solution.
   • If $\text{rank}(A)<n$ → free variables exist → infinitely many non-trivial solutions.
4. Express leading variables in terms of free variables.

Example

Solve the homogeneous system: $x+y-z=0,2x-y+z=0,x-2y+2z=0$

Augmented matrix: $\left[\begin{array}{ccccc}1& 1& -1& |& 0\\ 2& -1& 1& |& 0\\ 1& -2& 2& |& 0\end{array}\right]$

$R_2\to R_2-2R_1$, $R_3\to R_3-R_1$: $\left[\begin{array}{ccccc}1& 1& -1& |& 0\\ 0& -3& 3& |& 0\\ 0& -3& 3& |& 0\end{array}\right]$

$R_3\to R_3-R_2$: $\left[\begin{array}{ccccc}1& 1& -1& |& 0\\ 0& -3& 3& |& 0\\ 0& 0& 0& |& 0\end{array}\right]$

Rank = 2 < 3 unknowns → one free variable. Let $z=t$:

• From $R_2$: $-3y+3t=0\Rightarrow y=t$
• From $R_1$: $x+t-t=0\Rightarrow x=0$

Solution: $x=0,y=t,z=t$ for any $t\in \mathbb{R}$.

Non-Homogeneous Systems: Matrix Inversion Method (SLO M-11-A-19)

For $AX=B$ with $|A|\ne 0$:

$X=A^{-1}B=\frac{1}{|A|}\text{adj}(A)\cdot B$

• Condition: $|A|\ne 0$ (matrix must be non-singular).
• If $|A|=0$, the inverse does not exist and this method cannot be applied.

Non-Homogeneous Systems: Cramer's Rule (SLO M-11-A-19)

For $AX=B$ with $|A|\ne 0$, each variable is found by:

$x_i=\frac{|A_i|}{|A|}$

where $A_i$ is the matrix obtained by replacing the $i$-th column of $A$ with the column vector $B$.

Cases in Cramer's Rule:

| $|A|$ | $|A_i|$ | Interpretation | |-------|---------|----------------| | $\ne 0$ | any | Unique solution | | $=0$ | all $=0$ | Infinitely many solutions or no solution | | $=0$ | some $\ne 0$ | Inconsistent (no solution) |

Consistent and Inconsistent Systems (SLO M-11-A-18)

Using the rank of the coefficient matrix $A$ and augmented matrix $[A|B]$: