---
title: "Consistency of a linear system"
subject: "Mathematics III"
chapter: 1
key: "semester_3/mathematics_iii:consistency-of-a-linear-system"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/consistency-of-a-linear-system
site: BCT Engineering
---

# Consistency of a linear system

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2083 Baishakh, 5 marks; 2082 Baishakh, 3+2 marks; 2075 Asoj, 5 marks (asked 3 times)
- Video: [Consistency of system of linear equations using rank method](https://www.youtube.com/watch?v=povS4KL3nR8) — maths voice
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/consistency-of-a-linear-system
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

Test the consistency of the system x − 2y + 2z = 4, 3x + y + 4z = 6, x + y + z = 1. If found consistent, solve the system completely.

## Short answer

Row reduce the augmented matrix: rank(A) = rank([A|B]) = 3, so the system is consistent. Back substitution yields the unique solution x = −2, y = 0, z = 3.

## Answer

A system AX = B is consistent if rank A = rank [A|B]. If this common rank equals the number of unknowns, the solution is unique.

The augmented matrix is

```text
[A|B] = [1 −2 2 | 4]
        [3  1 4 | 6]
        [1  1 1 | 1]
```

#### Row reduction
R₂ → R₂ − 3R₁, R₃ → R₃ − R₁:

```text
[1 −2  2 |  4]
[0  7 −2 | −6]
[0  3 −1 | −3]
```

R₂ → R₂ − 2R₃:

```text
[1 −2  2 |  4]
[0  1  0 |  0]
[0  3 −1 | −3]
```

R₃ → R₃ − 3R₂:

```text
[1 −2  2 |  4]
[0  1  0 |  0]
[0  0 −1 | −3]
```

#### Consistency
rank A = 3 and rank [A|B] = 3 (three non-zero rows). Since the ranks are equal, the system is **consistent**. The rank equals the number of unknowns (3), so the solution is **unique**.

#### Solution (back substitution)
- −z = −3 ⇒ z = 3
- y = 0
- x − 2(0) + 2(3) = 4 ⇒ x = −2

**Check:** 3(−2) + 0 + 4(3) = 6 ✓; −2 + 0 + 3 = 1 ✓.

**Answer: x = −2, y = 0, z = 3.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. When is AX = B consistent?
   A. rank A < rank [A|B]
   B. rank A = rank [A|B]
   C. |A| = 0
   D. rank [A|B] = 0
   Answer: B. Consistency needs equal ranks of the coefficient and augmented matrices. (Key point: Consistency criterion)

2. Here rank A = rank [A|B] = 3 with 3 unknowns. What follows?
   A. Unique solution
   B. Infinitely many solutions
   C. No solution
   D. Exactly two solutions
   Answer: A. Common rank equal to the number of unknowns gives a unique solution. (Key point: Number of solutions)

3. What is the solution of the system?
   A. x = 2, y = 0, z = -3
   B. x = -2, y = 3, z = 0
   C. x = -2, y = 0, z = 3
   D. x = 0, y = -2, z = 3
   Answer: C. Back substitution: z = 3, y = 0, x = 4 - 6 = -2. (Key point: Solution)

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