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NDA Determinants

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This page covers NDA Determinants with complete concept notes, 42 graded practice MCQs, key points and exam-specific tips. Free to study.

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Concept Notes

Determinants— Rules & Concept

1.Core Concept

Read this first — the foundation of the topic

A determinant is a special number calculated from a square matrix. Think of it as a single value that tells us important properties about the matrix. For a 2x2 matrix, if elements are a, b, c, d (arranged as first row: a, b; second row: c, d), then determinant = ad - bc.

For 3x3 matrices, we use cofactor expansion.

2.Key Rules

Core rules you must know cold

(1) If any row or column has all zeros, determinant = 0. (2) Swapping two rows changes the sign of determinant. (3) If two rows are identical, determinant = 0. (4) Adding a multiple of one row to another row doesn't change the determinant. (5) Determinant of transpose equals original determinant.

3.Formula Block

Memorise — at least one formula appears in every paper
• 2x2 determinant: |A| = ad - bc
• Properties: |AB| = |A||B|, |A^T| = |A|, |kA| = k^n|A| for nxn matrix
• Area of triangle with vertices (x1,y1), (x2,y2), (x3,y3) = (1/2)|determinant of coordinate matrix|

• 3x3 determinant: Expand along any row or column using cofactors

4.Exam Patterns

What examiners ask — read before attempting PYQs

NDA frequently asks (1) Calculate 2x2 or 3x3 determinants, (2) Find area of triangle using determinants, (3) Properties-based questions, (4) Solve equations using Cramer's rule. Most questions are direct calculation type worth 2-4 marks.

5.Shortcuts

Use these to save 30–60 seconds per question

For 3x3 determinants, use Sarrus rule - write first two columns again on the right. Multiply diagonals going down-right (positive terms), multiply diagonals going down-left (negative terms). Add positive terms, subtract negative terms.

6.Worked Example

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

This is 2x2 matrix with a=3, b=1, c=2, d=4

2
Step 2

Apply formula |A| = ad - bc

3
Step 3

|A| = (3×4) - (1×2) = 12 - 2 = 10 Answer: 10 Worked Example 2: Find area of triangle with vertices A(1,2), B(3,4), C(5,1).

1
Step 1

Set up determinant matrix: |1 2 1; 3 4 1; 5 1 1|

2
Step 2

Expand along third column: 1×|3 4; 5 1| - 1×|1 2; 5 1| + 1×|1 2; 3 4|

3
Step 3

= 1×(3-20) - 1×(1-10) + 1×(4-6) = -17 + 9 - 2 = -10

4
Step 4

Area = (1/2)|determinant| = (1/2)×10 = 5 square units Answer: 5 square units

7.Exam Traps

Common mistakes students make — avoid these

Students often forget the sign changes when expanding determinants. When expanding along a row or column, the signs alternate starting with positive. The cofactor of element at position (i,j) has sign (-1)^(i+j).

Also, many students calculate 3x3 determinants incorrectly by mixing up the diagonal multiplication patterns.

Key Points to Remember

  • Determinant of 2x2 matrix with elements a,b,c,d is ad-bc
  • For 3x3 determinants, expand along any row or column using cofactors
  • Sarrus rule shortcut: extend first two columns, multiply diagonals down-right minus diagonals down-left
  • If any row or column is all zeros, determinant equals zero
  • Swapping two rows changes the sign of the determinant
  • Two identical rows make determinant equal to zero
  • Area of triangle = (1/2) × |determinant of coordinate matrix|
  • Property: |AB| = |A| × |B| for square matrices
  • Determinant of transpose matrix equals original determinant
  • Signs in cofactor expansion alternate: use (-1)^(i+j) for position (i,j)

Exam-Specific Tips

  • Determinant of identity matrix always equals 1
  • Determinant of zero matrix always equals 0
  • For nxn matrix A multiplied by scalar k: |kA| = k^n × |A|
  • Cramer's rule works only when system determinant is non-zero
  • Matrix is invertible if and only if its determinant is non-zero
  • Determinant of upper triangular matrix equals product of diagonal elements
  • Determinant of orthogonal matrix is either +1 or -1
  • For 2x2 matrix: inverse exists when ad-bc ≠ 0
Practice MCQs

Determinants — Practice Questions

42graded MCQs · easy to hard · full solution & trap analysis · showing 20 of 42

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Practice 1easy

If the determinant of a 2×2 matrix A = [[a, b], [c, d]] is 7, then the determinant of adj(A) is:

Practice 2easy

If A and B are 3×3 matrices with det(A) = 4 and det(B) = −2, then det(AB) is equal to:

Practice 3easy

If A is a 2×2 matrix with det(A) = 5, then det(A⁻¹) is equal to:

Practice 4easy

If A = [[2, 0], [0, 3]] and B = [[1, 1], [0, 1]], then det(AB) is equal to:

Practice 5easy

If A is a 2×2 matrix with det(A) = 3, then det(A⁻¹) is equal to:

Practice 6easy

If A = [[1, 2, 3], [0, 4, 5], [0, 0, 6]] is an upper triangular matrix, then det(A) is:

Practice 7easy

If A is a square matrix of order 2 and det(A) = 3, then det(A · adj(A)) is equal to:

Practice 8easy

If the determinant of a 3×3 matrix A is 6, and B is obtained by swapping two rows of A, then det(B) is:

Practice 9easy

If A is a 3×3 matrix such that det(A) = 5, then det(2A) is equal to:

Practice 10easy

Let A = [[1, 2], [3, 4]]. If B is the matrix obtained by interchanging the rows of A, then det(B) is:

Practice 11easy

If A = [[2, 1], [1, 2]] and B = [[1, 0], [0, 1]], then det(A − B) is equal to:

Practice 12easy

If A is a 3×3 matrix and det(A) = 0, which of the following statements is necessarily true?

Practice 13easy

Let A = [[1, 2], [3, 4]]. If B is a matrix such that det(B) = 2, then det(A²B) is equal to:

Practice 14medium

Let A = [[2, 1], [3, 4]] and B = [[1, 0], [0, 1]]. If det(A − λB) = 0, then the sum of all possible values of λ is:

Practice 15medium

Let A be a 3×3 matrix such that det(A) = 5. If B is the matrix obtained by interchanging the first and second rows of A, and C is the matrix obtained by multiplying the third row of A by 3, then det(B) + det(C) equals:

Practice 16medium

If A is a 3×3 matrix with det(A) = 4, then det(2A) equals:

Practice 17medium

Let A = [1, 2; 3, 4] (a 2×2 matrix). If B is obtained by adding 2 times the first row of A to the second row, then det(B) − det(A) equals:

Practice 18medium

Let A be a 3×3 matrix such that det(A) = 6. If A^T denotes the transpose of A, then det(A^T) + det(A) equals:

Practice 19medium

Let A and B be 3×3 matrices with det(A) = 3 and det(B) = −2. If C = AB (matrix product), then det(C) equals:

Practice 20medium

Let A be a 3×3 matrix such that det(A) = 5. If B is the matrix obtained by interchanging the first and third rows of A, then det(B) is:

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60-Second Revision — Determinants

  • Formula: 2x2 determinant = ad - bc, memorize this pattern
  • Remember: Sarrus rule for 3x3 - extend two columns, diagonal products
  • Trap: Sign alternation in cofactor expansion starts positive at (1,1)
  • Quick check: If any row/column is zeros, answer is zero
  • Area formula: Triangle area = (1/2)|coordinate determinant|
  • Property: Swapping rows changes sign, identical rows give zero
  • Remember: |AB| = |A||B| and |A^T| = |A|
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