# Systems of Linear Equations and Gaussian Elimination

Physics I · Linear Algebra and Geometry · https://tryals.app/learn/physics-i/systems-of-linear-equations-and-gaussian-elimination

## Linear Systems & Gaussian Form

A **system of linear equations** in matrix form is written as $A\vec{x} = \vec{b}$, where $A$ is the coefficient matrix, $\vec{x}$ is the variable vector, and $\vec{b}$ is the constant vector.

**Gaussian elimination** solves this by reducing the augmented matrix $[A|\vec{b}]$ into row echelon form using three safe operations:

| Operation | Notation | Rule |
| --- | --- | --- |
| **Swap** | $R_i \leftrightarrow R_j$ | Exchange rows |
| **Scale** | $R_i \to \alpha R_i$ | $\alpha \neq 0$ |
| **Add** | $R_i \to R_i + \alpha R_j$ | Add multiple |

**Example**: Solve a $2\times 2$ system:
$$\begin{pmatrix} 2 & 1 & | & 5 \\ 4 & -1 & | & 1 \end{pmatrix} \xrightarrow{R_2 - 2R_1} \begin{pmatrix} 2 & 1 & | & 5 \\ 0 & -3 & | & -9 \end{pmatrix}$$

Back-substitution yields $y = 3$ and $x = 1$. Row operations preserve the solution set, but yield a different matrix telling the same story.

## Solution Types & Ranks

Always check the rank of both $A$ and the augmented matrix $[A|\vec{b}]$ before interpreting your final system outcome.

| Solution Type | Rank Condition | Meaning |
| --- | --- | --- |
| **Unique** | $\mathrm{rk}(A) = \mathrm{rk}([A|\vec{b}]) = n$ | Exact fit |
| **Infinite** | $\mathrm{rk}(A) = \mathrm{rk}([A|\vec{b}]) < n$ | Free variables |
| **None** | $\mathrm{rk}(A) < \mathrm{rk}([A|\vec{b}])$ | Contradiction |

**Common pitfall:** A resulting row of the form $0\;0\;|\;c$ where $c \neq 0$ instantly means no solution. Stop immediately when this contradiction appears.

## Practice questions

5 of this lesson's 13 practice questions, with answers. The full set is in the app.

### 1. Order the steps of Gaussian elimination.

**Answer:**

1. Write the augmented matrix [A|b]
2. Use row operations to get upper triangular form
3. Identify pivots and free variables
4. Back-substitute to find the solution

**Why:** Gaussian elimination reduces the system step by step to an easily solvable form.

Page: https://tryals.app/practice/physics-i/systems-of-linear-equations-and-gaussian-elimination/order-the-steps-of-gaussian-elimination

### 2. While solving a linear system you reach the row $\;0\;\;0\;\;|\;\,5$. What has the algebra just told you?

A. One variable equals 5
B. There are infinitely many solutions
C. A row swap is needed to continue
D. The system is inconsistent

**Answer:** D. The system is inconsistent

**Why:** The row asserts $0 = 5$, a contradiction, so the original equations demand the impossible (geometrically: parallel lines that never meet). Elimination doesn’t just find solutions; it delivers verdicts about whether solutions exist at all.

Page: https://tryals.app/practice/physics-i/systems-of-linear-equations-and-gaussian-elimination/while-solving-a-linear-system-you-reach-the-row-0-0-5-what-has-the

### 3. A system of 3 equations in 3 unknowns always has exactly one solution.

**Answer:** False

**Why:** It may have zero, one, or infinitely many solutions depending on the rank.

Page: https://tryals.app/practice/physics-i/systems-of-linear-equations-and-gaussian-elimination/a-system-of-3-equations-in-3-unknowns-always-has-exactly-one-solution

### 4. Gaussian elimination changes a system’s appearance completely while leaving its solutions untouched. What makes that legitimate?

A. The operations are applied to both sides of every equation equally
B. The determinant is preserved by each operation
C. Every row operation is reversible, so the solution set is kept
D. The solutions are recomputed after each step to keep them consistent

**Answer:** C. Every row operation is reversible, so the solution set is kept

**Why:** Reversibility is what licenses the method. An irreversible operation, multiplying a row by zero, is excluded precisely because it would add solutions the original system did not have.

Page: https://tryals.app/practice/physics-i/systems-of-linear-equations-and-gaussian-elimination/gaussian-elimination-changes-a-systems-appearance-completely-while

### 5. Which properties of a matrix survive **every** elementary row operation?

A. The determinant
B. The solution set of $Ax = 0$
C. The rank
D. The row space

**Answer:** B. The solution set of $Ax = 0$; C. The rank; D. The row space

**Why:** Rank, row space and the null space are what row reduction is built to preserve, which is why solving $Ax = 0$ by elimination is legitimate. The determinant is not: a row swap flips its sign and scaling a row by $k$ multiplies it by $k$.

Page: https://tryals.app/practice/physics-i/systems-of-linear-equations-and-gaussian-elimination/which-properties-of-a-matrix-survive-every-elementary-row-operation