# Gaussian Elimination and Rank

Mathematics I · Matrices and Vectors · https://tryals.app/learn/mathematics-i/gaussian-elimination-and-rank

## Skeleton of a System

**Gaussian elimination** transforms a matrix into **row echelon form**: a staircase pattern where each row's leading nonzero entry (**pivot**) sits to the right of the one above it.

Applying **row operations** (swapping, scaling, adding multiples) preserves solutions while clearing entries below pivots. **Reduced** row echelon form makes every pivot 1 and the sole nonzero entry in its column.

| Operation | What it does | Effect on Solutions |
| --- | --- | --- |
| Swap | Interchange two rows | Preserved |
| Scale | Multiply a row by $c \neq 0$ | Preserved |
| Add | Add $c \times$ row to another | Preserved |

> **Pitfall:** A **zero row** ($0=0$) means redundancy and infinitely many solutions. An **inconsistent row** ($0=1$) means contradiction and **no solution**.

## Rank and Solution Count

The number of pivots is the **rank**: the count of independent rows. Rank measures genuine information:

- **Full rank** ($\text{rank} = n$ for $n \times n$): Invertible, unique solution to $A\mathbf{x} = \mathbf{b}$.
- **Rank deficiency** ($\text{rank} < n$): Dependent rows, free variables, no unique solution.

The number of **free variables** equals (unknowns $-$ rank). Zero free variables yield a single solution; more yield infinite.

$$\text{Free variables} = n - \text{rank}$$

## Practice questions

6 of this lesson's 9 practice questions, with answers. The full set is in the app.

### 1. A row reading $0\ 0\ 0 \mid 5$ after elimination means the system has infinitely many solutions.

**Answer:** False

**Why:** False. That row asserts $0 = 5$, so the system is *inconsistent* and has no solution at all. The row that signals infinitely many is $0\ 0\ 0 \mid 0$, no information, leaving a free variable. The two look alike and mean opposite things.

Page: https://tryals.app/practice/mathematics-i/gaussian-elimination-and-rank/a-row-reading-0-0-0-5-after-elimination-means-the-system-has

### 2. Gaussian elimination is taught with fractions and exact arithmetic. Real implementations swap rows to put the largest entry on the pivot first. What does that **partial pivoting** protect against?

A. Changing the determinant's sign, which row swaps would otherwise do
B. Dividing by a very small pivot, which magnifies rounding error
C. Producing fractions, which are slower to compute than decimals
D. Encountering a zero pivot, which is the only case where elimination fails

**Answer:** B. Dividing by a very small pivot, which magnifies rounding error

**Why:** Dividing by a tiny pivot amplifies whatever error is already there, so a mathematically irrelevant choice becomes numerically decisive. It is a clean example of the gap between an algorithm and its implementation, the algebra is indifferent to row order and the floating-point arithmetic is not.

Page: https://tryals.app/practice/mathematics-i/gaussian-elimination-and-rank/gaussian-elimination-is-taught-with-fractions-and-exact-arithmetic

### 3. Order the steps of running Gaussian elimination on a 3 by 3 coefficient matrix.

**Answer:**

1. Choose a nonzero entry in the first column as the pivot, swapping rows if necessary
2. Add multiples of the pivot row to the rows below to clear the rest of the first column
3. Move to the second column and repeat the process below the first pivot
4. Stop once the matrix is in row echelon form with zero rows at the bottom
5. Count the pivots to read off the rank

**Why:** Pivot, clear below, move right, repeat, then count. Clearing only below the pivot is what makes the staircase, and the pivot count at the end is the rank, the same number that governs invertibility and the count of free variables.

Page: https://tryals.app/practice/mathematics-i/gaussian-elimination-and-rank/order-the-steps-of-running-gaussian-elimination-on-a-3-by-3

### 4. A row of zeros appears in the coefficient part during elimination. Why does that not by itself mean the system has no solution?

A. Because a zero row means the matrix was not reduced correctly
B. Because zero rows indicate the system has exactly one solution
C. Because everything depends on the augmented entry on the right
D. Because zero rows are always redundant and may simply be deleted

**Answer:** C. Because everything depends on the augmented entry on the right

**Why:** The zero row alone says nothing; the augmented column decides. $0 = 0$ is redundancy and $0 = 5$ is inconsistency, the same left-hand side with opposite verdicts.

Page: https://tryals.app/practice/mathematics-i/gaussian-elimination-and-rank/a-row-of-zeros-appears-in-the-coefficient-part-during-elimination

### 5. A system in 4 unknowns reduces to 3 pivots with no inconsistent row. Select every TRUE statement.

A. The system has a unique solution
B. The system is inconsistent
C. The rank is 3
D. The system has infinitely many solutions
E. There is exactly 1 free variable

**Answer:** C. The rank is 3; D. The system has infinitely many solutions; E. There is exactly 1 free variable

**Why:** 3 pivots means rank 3, so 4 - 3 = 1 free variable, and one free parameter already produces infinitely many solutions. Uniqueness would need 4 pivots, and inconsistency was ruled out by hypothesis. Rank alone settles all three questions.

Page: https://tryals.app/practice/mathematics-i/gaussian-elimination-and-rank/a-system-in-4-unknowns-reduces-to-3-pivots-with-no-inconsistent-row

### 6. Sort each matrix by whether it is in row echelon form. Each row's leading entry must sit strictly to the right of the one above, and zero rows must sink to the bottom.

**Answer:**

- In row echelon form: rows (1, 2, 3), (0, 1, 4), (0, 0, 5), rows (1, 0, 0), (0, 1, 0), (0, 0, 1), rows (2, 1), (0, 0)
- Not in row echelon form: rows (1, 2, 3), (0, 0, 0), (0, 1, 4), rows (0, 1, 2), (1, 0, 3)

**Why:** m1, m4 and m5 have leading entries marching strictly rightward with any zero row at the bottom. m2 places a zero row above a nonzero one. In m3 the second row's leading entry is in column 1, to the LEFT of the first row's in column 2, a swap would fix it. Note that pivots need not equal 1 for plain echelon form.

Page: https://tryals.app/practice/mathematics-i/gaussian-elimination-and-rank/sort-each-matrix-by-whether-it-is-in-row-echelon-form-each-rows