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Determinants – Class 12 Mathematics notes, important questions and MCQs

NCERT Class 12 Mathematics, chapter 4. Notes, key terms, important questions with answers and MCQs.

Chapter overview

A determinant is a number linked with a square matrix. It gives areas of triangles, tests invertibility, and solves linear systems by the matrix method.

Evaluation and properties

|A| for 2 × 2: ad − bc; 3 × 3 by expanding along a row or column with cofactors Aᵢⱼ = (−1)^(i+j)Mᵢⱼ.
Interchanging two rows changes the sign; two identical rows give 0; |kA| = kⁿ|A| for order n.
|AB| = |A||B|; |A′| = |A|.

Inverse and applications

A⁻¹ = adj A / |A| (exists only if |A| ≠ 0); A(adj A) = |A|I; |adj A| = |A|ⁿ⁻¹.
System AX = B: unique solution X = A⁻¹B if |A| ≠ 0.
Area of triangle = ½ |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|; zero area ⇒ collinear points.

Key terms

Singular matrix: square matrix with determinant zero
Adjoint: transpose of the cofactor matrix
Cofactor: Aᵢⱼ = (−1)^(i+j) Mᵢⱼ, where Mᵢⱼ is the minor obtained by deleting row i and column j

Formulas

A⁻¹ = adj A / |A|
|kA| = kⁿ|A|
|adj A| = |A|ⁿ⁻¹

Important questions with answers

Find x if |2 4; 5 1| = |2x 4; 6 x|.2 marks

2 − 20 = 2x² − 24, so 2x² = 6, x = ±√3.

Using determinants, check whether (1, 5), (2, 3) and (−2, −11) are collinear.3 marks

½|1(3 + 11) + 2(−11 − 5) + (−2)(5 − 3)| = ½|14 − 32 − 4| = 11 ≠ 0. Not collinear.

Solve by the matrix method: x + y + z = 6, y + 3z = 11, x − 2y + z = 0.5 marks

A = [1 1 1; 0 1 3; 1 −2 1], B = [6; 11; 0]. |A| = 1(1 + 6) − 1(0 − 3) + 1(0 − 1) = 7 + 3 − 1 = 9 ≠ 0, so a unique solution exists. Cofactors give adj A = [7 −3 2; 3 0 −3; −1 3 1]. X = A⁻¹B = (1/9)[42 − 33 + 0; 18 + 0 − 0; −6 + 33 + 0] = (1/9)[9; 18; 27] = [1; 2; 3]. So x = 1, y = 2, z = 3 (check: 1 + 2 + 3 = 6 ✓).

Quick revision

Singular ⇒ no inverse.
Swapping rows changes sign.
Collinear points give zero area.

Exam tips

If |A| = 0 and (adj A)B ≠ 0, the system is inconsistent.
Before expanding a 3 × 3 determinant, use row/column operations to create zeros; expand along the row with most zeros.

Practice MCQs

Value of |2 3; 1 4| is
1. 8
2. 11
3. 5
4. −5
Show answer
Answer: (c) 5. 8 − 3.
A matrix with determinant zero is
1. invertible
2. identity
3. singular
4. symmetric
Show answer
Answer: (c) singular. Singular.
If A is 3 × 3 and |A| = 4, then |2A| is
1. 8
2. 16
3. 32
4. 12
Show answer
Answer: (c) 32. 2³ × 4.
If two rows of a determinant are identical, its value is
1. 1
2. doubled
3. −1
4. 0
Show answer
Answer: (d) 0. Property.
If |A| = 3 for a 2 × 2 matrix, then |A⁻¹| is
1. 3
2. 9
3. 1/3
4. −3
Show answer
Answer: (c) 1/3. |A⁻¹| = 1/|A|.
If |A| = 5 for a 3 × 3 matrix, then |adj A| is
1. 5
2. 15
3. 125
4. 25
Show answer
Answer: (d) 25. |A|² = 25.
Area of the triangle with vertices (0, 0), (4, 0), (0, 3) is
1. 12
2. 3.5
3. 7
4. 6
Show answer
Answer: (d) 6. ½ × 4 × 3.
The value of |1 a b+c; 1 b c+a; 1 c a+b| is
1. abc
2. a + b + c
3. 0
4. 1
Show answer
Answer: (c) 0. C₃ → C₃ + C₂ gives two equal columns.

All chapters of Class 12 Mathematics

Written by Kundali IQ from standard, well-established facts to help with revision. This is not an official NCERT publication; always read your textbook too.