Systems of Linear Equations & Matrices: Augmented Forms, Consistency & Solutions

Core Concepts & Definitions

1. Systems of Linear Equations

A system of linear equations consists of m linear equations in n variables: $$\{\begin{array}{l}{a}_{11}{x}_1+a_{12}{x}_2+\cdots +a_{1n}{x}_n=b_1\\ a_{21}{x}_1+a_{22}{x}_2+\cdots +a_{2n}{x}_n=b_2\\ ⋮\\ a_{m1}{x}_1+a_{m2}{x}_2+\cdots +a_{mn}{x}_n=b_m\end{array}$$

• Solution: A vector $\mathbf{x}=(x_1,x_2,\dots ,x_n)$ that satisfies all equations simultaneously.
• Consistency: A system is consistent if it has at least one solution; otherwise, it is inconsistent.

Types of Solutions

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2. Augmented Matrix & Row Operations

Augmented Matrix

An augmented matrix combines the coefficient matrix $A$ and the constant vector $\mathbf{b}$ into a single matrix: $$[A|\mathbf{b}]=\left[\begin{array}{cccccc}{a}_{11}& a_{12}& \dots & a_{1n}& |& b_1\\ a_{21}& a_{22}& \dots & a_{2n}& |& b_2\\ ⋮& ⋮& \ddots & ⋮& |& ⋮\\ a_{m1}& a_{m2}& \dots & a_{mn}& |& b_m\end{array}\right]$$ Example: For the system: $$\{\begin{array}{l}x+6y+2z=-5\\ -x+3z=1\\ -y-2z=3\end{array}$$ The augmented matrix is: $$\left[\begin{array}{ccccc}1& 6& 2& |& -5\\ -1& 0& 3& |& 1\\ 0& -1& -2& |& 3\end{array}\right]$$

Elementary Row Operations (EROs)

Used to transform the augmented matrix into row-echelon form (REF) or reduced row-echelon form (RREF):

1. Row Swapping ($R_i\leftrightarrow R_j$): Exchange two rows.
2. Scalar Multiplication ($k{R}_i\to R_i$): Multiply a row by a non-zero scalar.
3. Row Addition ($R_i+k{R}_j\to R_i$): Add a multiple of one row to another.

Goal: Achieve a matrix where:

• All non-zero rows are above any rows of all zeros.
• The leading coefficient (pivot) of a non-zero row is always strictly to the right of the leading coefficient of the row above it.
• All entries in a column below a pivot are zeros.

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3. Solving Systems Using Gaussian Elimination

Steps:

1. Write the augmented matrix $[A|\mathbf{b}]$.
2. Perform EROs to convert it to REF.
3. Back-substitute to find solutions (or express in parametric form if free variables exist).

Worked Example: General Solution

Problem: Find the general solution of the system with augmented matrix: $$\left[\begin{array}{ccccc}1& 6& 2& |& -5\\ -1& 0& 3& |& 1\\ 0& -1& -2& |& 3\end{array}\right]$$

Solution:

1. Step 1: Add $R_1$ to $R_2$ to eliminate $-1$ in $R_2$: $$R_2\to R_2+R_1=[0,6,5|-4]$$ New matrix: $$\left[\begin{array}{ccccc}1& 6& 2& |& -5\\ 0& 6& 5& |& -4\\ 0& -1& -2& |& 3\end{array}\right]$$

2. Step 2: Make the pivot in $R_3$ equal to 1 by multiplying $R_3$ by $-1$: $$R_3\to -1\cdot R_3=[0,1,2|-3]$$ Matrix: $$\left[\begin{array}{ccccc}1& 6& 2& |& -5\\ 0& 6& 5& |& -4\\ 0& 1& 2& |& -3\end{array}\right]$$

3. Step 3: Eliminate the $6$ in $R_2$ using $R_3$: $$R_2\to R_2-6R_3=[0,0,-7|14]$$ Simplified matrix: $$\left[\begin{array}{ccccc}1& 6& 2& |& -5\\ 0& 0& -7& |& 14\\ 0& 1& 2& |& -3\end{array}\right]$$

4. Step 4: Divide $R_2$ by $-7$: $$R_2\to -\frac{1}{7}{R}_2=[0,0,1|-2]$$ Final REF: $$\left[\begin{array}{ccccc}1& 6& 2& |& -5\\ 0& 0& 1& |& -2\\ 0& 1& 2& |& -3\end{array}\right]$$

5. Back-Substitution:

   • From $R_2$: $z=-2$.
   • From $R_3$: $y+2(-2)=-3⟹y=1$.
   • From $R_1$: $x+6(1)+2(-2)=-5⟹x=-5$.

Solution: $(x,y,z)=(-5,1,-2)$ (unique solution).

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4. Homogeneous Systems

A homogeneous system has the form $A\mathbf{x}=0$, where $\mathbf{b}=0$.

Key Properties

• Always at least one solution: $\mathbf{x}=0$ (trivial solution).
• Non-trivial solutions exist if and only if the system is dependent (i.e., $\det (A)=0$ or rank$(A)<n$).

Example: $$\{\begin{array}{l}x-2y+3z=0\\ -x+2y-4z=0\end{array}$$

• The second equation is $-1$ times the first, so the system is dependent.
• Non-trivial solutions exist (e.g., $y=1,z=1⟹x=-1$).

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5. Consistency of Linear Systems

A system $A\mathbf{x}=\mathbf{b}$ is consistent if: $$\text{rank}(A)=\text{rank}([A|\mathbf{b}]).$$ Otherwise, it is inconsistent.

Worked Example: Consistency Check Problem: Determine if the system is consistent: $$\{\begin{array}{l}-2x_1-3x_2+4x_3=5\\ x_2-2x_3=4\\ x_1+3x_2-x_3=2\end{array}$$

Solution:

1. Write the augmented matrix: $$\left[\begin{array}{ccccc}-2& -3& 4& |& 5\\ 0& 1& -2& |& 4\\ 1& 3& -1& |& 2\end{array}\right]$$
2. Perform EROs:
   • Swap $R_1$ and $R_3$: $$\left[\begin{array}{ccccc}1& 3& -1& |& 2\\ 0& 1& -2& |& 4\\ -2& -3& 4& |& 5\end{array}\right]$$
   • Eliminate $-2$ in $R_3$: $$R_3\to R_3+2R_1=[0,3,2|9]$$
   • Eliminate $3$ in $R_3$ using $R_2$: $$R_3\to R_3-3R_2=[0,0,8|-3]$$
3. Final REF: $$\left[\begin{array}{ccccc}1& 3& -1& |& 2\\ 0& 1& -2& |& 4\\ 0& 0& 8& |& -3\end{array}\right]$$
   • $\text{rank}(A)=3$ (all rows are non-zero).
   • $\text{rank}([A|\mathbf{b}])=3$.
   • Conclusion: The system is consistent (unique solution exists).

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6. Applications of Linear Systems

1. Network Flow Problems: Modeling traffic or fluid dynamics.
2. Economics: Input-output models (e.g., Leontief models).
3. Computer Graphics: Homogeneous coordinates for transformations.
4. Engineering: Circuit analysis (Kirchhoff’s laws).
5. Statistics: Least squares fitting (covered in Unit 7).

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7. Comparison: Direct vs. Iterative Methods

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Exam Tip

1. Augmented Matrix Questions:

   • Always perform EROs step-by-step and show intermediate matrices.
   • For general solutions, express free variables explicitly (e.g., $x=\cdots +s\cdot \mathbf{v}$).

2. Consistency Checks:

   • Compare $\text{rank}(A)$ and $\text{rank}([A|\mathbf{b}])$.
   • If ranks differ, the system is inconsistent.

3. Homogeneous Systems:

   • Remember: Non-trivial solutions exist only if $\det (A)=0$ or rank$(A)<n$.

4. Common Mistakes to Avoid:

   • Forgetting to perform row operations systematically (e.g., skipping normalization).
   • Misinterpreting REF vs. RREF (RREF has pivots = 1 and zeros above/below).
   • Incorrectly identifying free variables in parametric solutions.

5. Past Exam Patterns:

   • Define + Solve: Expect 5–7 marks for definitions (e.g., augmented matrix) + 8–10 marks for solving.
   • Consistency: Always verify ranks explicitly.
   • Applications: Relate to real-world scenarios (e.g., "Why is Gaussian elimination used in circuit analysis?").

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Practice Problems:

1. Solve the system with augmented matrix: $$\left[\begin{array}{ccccc}1& 2& -1& |& 3\\ 2& 3& 1& |& 8\\ 1& 1& 1& |& 4\end{array}\right]$$
2. Determine if the system $x+y+z=1$, $2x-y+3z=4$, $x+4y-z=2$ is consistent.
3. Find the general solution of the homogeneous system: $$\{\begin{array}{l}x+2y-z=0\\ 2x+4y-2z=0\\ x-y+z=0\end{array}$$