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NDA System of Linear Equations

Study Material — 1 PYQs (2020–2020) · Concept Notes · Shortcuts

NDA System of Linear Equations is a frequently tested subtopic — 1 previous year questions from 2020–2020 papers are included below with concept notes, key rules and shortcut tricks.

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Previous Year Questions

NDA System of Linear Equations — Past Exam Questions

1 questions from actual NDA papers · all shown free · click option to reveal solution

Exam Q 12020Previous Year Pattern

The system of linear equations: 2x + 3y + z = 5 4x + 6y + 2z = 10 3x + 5y + z = 8 has:

Concept Notes

System of Linear Equations— Rules & Concept

1.Core Concept

Read this first — the foundation of the topic

SYSTEM OF LINEAR EQUATIONS — MATRICES & DETERMINANTS ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

CORE CONCEPT ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

A system of linear equations is a group of two or more equations with the same set of unknowns (like x, y, z). We want to find values of x, y, z that satisfy ALL equations at the same time. Using matrices, we can solve these systems quickly and systematically. The standard form of 3 equations in 3 unknowns:

a1x + b1y + c1z = d1 a2x + b2y + c2z = d2

a3x + b3y + c3z = d3 This is written in matrix form as: AX = B

Where A is the coefficient matrix, X is the column matrix of unknowns, and B is the column matrix of constants. ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

KEY RULES — CONSISTENCY OF THE SYSTEM ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

The most tested concept is whether the system is CONSISTENT (has solution) or INCONSISTENT (no solution). Case 1 — det(A) ≠ 0:

The system has a UNIQUE solution. Use Cramer's Rule or inverse matrix method. Case 2 — det(A) = 0 AND the augmented matrix [A|B] has the same rank as A:

The system has INFINITELY MANY solutions (dependent equations). Case 3 — det(A) = 0 AND rank of [A|B] > rank of A:

The system has NO solution (inconsistent). ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

2.Formula Block

Memorise — at least one formula appears in every paper
For AX = B, if D = det(A) ≠ 0:
x = D1/D, y = D2/D, z = D3/D
D1 = Replace column 1 of A with B, then find determinant.
D2 = Replace column 2 of A with B, then find determinant.
D3 = Replace column 3 of A with B, then find determinant.

CRAMER'S RULE

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

3.Exam Patterns

What examiners ask — read before attempting PYQs

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ NDA most commonly asks: 1. Is the given system consistent or inconsistent? 2. Find the solution using Cramer's Rule. 3. For what value of k does the system have no solution or infinite solutions? 4.

Homogeneous systems — does it have only trivial solution or non-trivial? For HOMOGENEOUS system (B = 0): AX = 0 — If det(A) ≠ 0 → Only trivial solution (x=y=z=0) — If det(A) = 0 → Non-trivial (infinite) solutions exist ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ SHORTCUT / TRICK ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ TRICK 1 — The Determinant Shortcut: Always calculate det(A) FIRST. If det(A) ≠ 0, directly write 'unique solution exists' and use Cramer's Rule. Do not waste time checking rank. TRICK 2 — For 'value of k' type questions: Set det(A) = 0 and solve for k. This gives you the critical value.

Then check one equation from the system to decide if it's infinite solutions or no solution. ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

4.Worked Example

Solve this step-by-step before moving on
1
Step 1

Write coefficient matrix A: A = |2 1| |1 3|

2
Step 2

Find D = det(A) D = (2×3) − (1×1) = 6 − 1 = 5

3
Step 3

Find D1 (replace column 1 with B = [5, 10]) D1 = |5 1| |10 3| D1 = (5×3) − (1×10) = 15 − 10 = 5

4
Step 4

Find D2 (replace column 2 with B) D2 = |2 5| |1 10| D2 = (2×10) − (5×1) = 20 − 5 = 15

5
Step 5

Apply Cramer's Rule x = D1/D = 5/5 = 1 y = D2/D = 15/5 = 3 Answer: x = 1, y = 3 Verification: 2(1) + 3 = 5 ✓ and 1 + 3(3) = 10 ✓ ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

5.Exam Traps

Common mistakes students make — avoid these

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ Students confuse det(A) = 0 with 'no solution'. Remember — det(A) = 0 means EITHER no solution OR infinite solutions. You MUST check the augmented matrix [A|B] to decide which one.

This distinction is heavily tested in NDA.

Key Points to Remember

  • Matrix form of AX = B: A is coefficient matrix, X is unknowns matrix, B is constants matrix.
  • If det(A) ≠ 0, the system has a unique solution — use Cramer's Rule directly.
  • If det(A) = 0 and rank[A|B] = rank[A], the system has infinitely many solutions.
  • If det(A) = 0 and rank[A|B] > rank[A], the system is inconsistent (no solution).
  • Cramer's Rule: x = D1/D, y = D2/D, z = D3/D where D = det(A).
  • For homogeneous system AX = 0: det(A) = 0 gives non-trivial solutions; det(A) ≠ 0 gives only trivial solution (0,0,0).
  • To find D1, replace the FIRST column of A with column matrix B and compute determinant.
  • det(A) = 0 alone does NOT confirm no solution — always verify with augmented matrix rank.

Exam-Specific Tips

  • Cramer's Rule is applicable ONLY when det(A) ≠ 0 (non-singular matrix).
  • A homogeneous system AX = 0 always has at least one solution: the trivial solution x = y = z = 0.
  • For a non-homogeneous system, det(A) = 0 leads to either no solution or infinitely many solutions — never a unique solution.
  • The augmented matrix is written as [A|B], formed by appending the constant matrix B to the coefficient matrix A.
  • A system of n equations in n unknowns has a unique solution if and only if the coefficient matrix is non-singular (det ≠ 0).
  • Rank of a matrix = number of non-zero rows in its row echelon form; for unique solution, rank(A) = rank[A|B] = number of unknowns.
  • In NDA exams, 'for what value of k does the system have no unique solution' always means: set det(A) = 0 and solve for k.
Practice MCQs

System of Linear Equations — Practice Questions

46graded MCQs · easy to hard · full solution & trap analysis · showing 20 of 46

All MCQs →
Practice 1easy

The system of equations: 2x + y + z = 3 x + 2y + z = 4 x + y + 2z = 5 has the solution (x, y, z) equal to:

Practice 2easy

The system of linear equations 2x + 3y = 5 4x + 6y = 10 has how many solutions?

Practice 3easy

Using Cramer's rule, find the value of x in the system: x + 2y = 5 3x − y = 4

Practice 4easy

For the system of equations 2x + y + z = 3 x − y + 2z = 5 3x + 2y − z = 2 the coefficient matrix A has determinant det(A) = −15. Using the matrix method, which statement is correct?

Practice 5easy

The system of linear equations x + 2y = 3 2x + 4y = 5 has which of the following properties?

Practice 6easy

If the system of equations ax + by = 1 cx + dy = 2 has the unique solution x = 1, y = 1, then which of the following must be true?

Practice 7easy

The system of linear equations: 2x + 3y = 5 4x + 6y = 10 has how many solutions?

Practice 8easy

The system of equations: ax + by = 1 cx + dy = 1 has a unique solution if and only if:

Practice 9easy

If the system of equations: px + qy = 1 rx + sy = 1 has no solution, then which of the following must be true?

Practice 10easy

The system of linear equations: 2x + 3y = 5 4x + 6y = 10 has:

Practice 11easy

If the system: 2x + 3y + z = 6 4x + 6y + 2z = 12 x + y + z = 5 has a solution, what is the rank of the augmented matrix [A|B]?

Practice 12easy

Using Cramer's rule, find the value of x in the system: x + 2y = 5 3x - y = 4

Practice 13easy

For the system of equations: ax + by = 1 cx + dy = 1 to have a unique solution, which condition must be satisfied?

Practice 14easy

For what value of k does the system: x + 2y = 3 2x + 4y = k have infinitely many solutions?

Practice 15medium

The system of equations: ax + by = 1 cx + dy = 1 has a unique solution if and only if which condition holds?

Practice 16medium

For the system of equations: ax + by = 4 cx + dy = 5 to have a unique solution, which of the following must be true?

Practice 17medium

Consider the system of linear equations: 2x + 3y - z = 5 4x + 6y - 2z = 10 x - y + 2z = 3 Which of the following statements is correct?

Practice 18medium

Using Cramer's rule, find the value of y in the system: 3x + 2y = 7 5x - y = 4 Which of the following is correct?

Practice 19medium

Consider the matrix equation AX = B, where: A = [1 2] [3 4] B = [5] [11] If X = [x], find x + y using the matrix method (inverse method). [y] Which of the following is correct?

Practice 20medium

The system of linear equations: 2x + 3y + 4z = 5 4x + 6y + 8z = 10 3x + 5y + 7z = 9 has how many solutions?

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60-Second Revision — System of Linear Equations

  • Remember: det(A) ≠ 0 → Unique solution. Use Cramer's Rule: x = D1/D, y = D2/D, z = D3/D.
  • Remember: det(A) = 0 does NOT automatically mean no solution — check rank of augmented matrix [A|B].
  • Formula: D1 = replace column 1 with B; D2 = replace column 2 with B; D3 = replace column 3 with B.
  • Trap: For homogeneous system (B=0), det(A) = 0 means infinite (non-trivial) solutions — NOT inconsistent.
  • Trick: In 'find k for no unique solution' questions — always set det(A) = 0 and solve. Quick 1-mark win.
  • Remember: Consistent system = has at least one solution. Inconsistent = no solution at all.
  • Trap: Never apply Cramer's Rule when D = 0 — it breaks down completely. Flag such systems using rank method.
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