Linear System of Equations (Consistency & Inconsistency) – Engineering Maths

A system of linear equations is a fundamental topic in Linear Algebra under Engineering Mathematics. Questions based on consistency and inconsistency are frequently asked in GATE, ESE, AE/JE exams.

Basic Concept

A system of linear equations can be written in matrix form as:

AX = B

Where:

A = Coefficient matrix
X = Variable matrix
B = Constant matrix

Augmented Matrix

The augmented matrix is formed by combining A and B:

[A | B]

Consistency & Inconsistency

1. Consistent System

A system is consistent if it has at least one solution.

2. Inconsistent System

A system is inconsistent if it has no solution.

Rank Conditions (VERY IMPORTANT)

Let:

Rank(A) = rank of coefficient matrix
Rank([A|B]) = rank of augmented matrix
n = number of variables

Condition	Result
Rank(A) = Rank([A\|B]) = n	Unique Solution
Rank(A) = Rank([A\|B]) < n	Infinite Solutions
Rank(A) ≠ Rank([A\|B])	No Solution (Inconsistent)

Solved Examples

Example 1 (Unique Solution)

System:

1	1	\|	2
2	3	\|	5

Rank(A) = 2, Rank([A|B]) = 2, n = 2

Answer: Unique Solution (Consistent)

Example 2 (Infinite Solutions)

1	2	\|	3
2	4	\|	6

Second row = 2 × first row

Rank(A) = 1, Rank([A|B]) = 1 < n

Answer: Infinite Solutions

Example 3 (Inconsistent System)

1	2	\|	3
2	4	\|	7

Left side proportional, right side NOT proportional

Rank(A) = 1, Rank([A|B]) = 2

Answer: No Solution (Inconsistent)

Example 4

If Rank(A) = Rank([A|B]) = 3 and n = 4

Answer: Infinite Solutions

Shortcut Tricks

Always compare Rank(A) and Rank([A|B])
Check proportional rows quickly
If RHS differs → Inconsistent
Count pivots after row reduction

Questions

Q1. When does system have no solution?

Answer: Rank(A) ≠ Rank([A|B])

Q2. When does system have infinite solutions?

Answer: Rank(A) = Rank([A|B]) < n

Exam Tips

Memorize rank conditions clearly
Avoid solving full equations
Use row reduction smartly
Focus on conceptual clarity

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