1st Year Math Chapter 4 Exercise 4.2 Notes: Determinants, Adjoint, and Inverse of a Matrix (Punjab Board 2026-27)

Complete concept notes, formulas, and solved examples for ICS/FSc Part 1 Mathematics, Punjab Board (PECTAA), Single National Curriculum 2026-27 session.

Exercise 4.2 builds on the matrix basics from Exercise 4.1, introducing the determinant of a matrix, minors and cofactors, and how to find a matrix’s adjoint and inverse.

What Does This Exercise Cover?

This exercise focuses on a single number — the determinant — that reveals whether a matrix has an inverse, and on the step-by-step process for actually finding that inverse when it exists.

Key Concepts

1. Determinant of a 2×2 Matrix

For A = [[a, b], [c, d]], the determinant is det(A) = ad − bc. It is a single number computed from the matrix’s entries.

2. Minor and Cofactor

For larger matrices, the minor Mᵢⱼ of an entry is the determinant of the smaller matrix left after removing its row i and column j. The cofactor Cᵢⱼ is the minor multiplied by (−1)^(i+j), which simply adjusts its sign based on position.

3. Singular and Non-Singular Matrices

A square matrix is called singular if its determinant equals zero, and non-singular if its determinant is not zero. Only non-singular matrices have an inverse.

4. Adjoint of a Matrix

The adjoint of a matrix is the transpose of its cofactor matrix (the matrix formed by replacing every entry with its cofactor, then flipping rows and columns).

5. Inverse of a Matrix

The inverse of a non-singular matrix A is given by A⁻¹ = (1/det(A)) × adj(A). Since this involves dividing by det(A), the inverse only exists when det(A) ≠ 0.

2×2 Matrix Formulas (for A = [[a, b], [c, d]])

Solved Examples

Example 1: Find the determinant of A = [[3, 4], [2, 5]].

det(A) = (3 × 5) − (4 × 2) = 15 − 8.

Answer: det(A) = 7

Example 2: Find the inverse of A = [[2, 1], [5, 3]].

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

adj(A) = [[3, −1], [−5, 2]].

Since det(A) = 1, A⁻¹ = (1/1) × adj(A).

Answer: A⁻¹ = [[3, −1], [−5, 2]]

Example 3: Determine whether A = [[2, 4], [1, 2]] is singular or non-singular.

det(A) = (2 × 2) − (4 × 1) = 4 − 4 = 0.

Answer: det(A) = 0, so A is singular and has no inverse

Sample MCQs

1. The determinant of [[a, b], [c, d]] is:

a) ac − bd   b) ad − bc   c) ab − cd   d) ad + bc

Answer: b) ad − bc

2. A matrix is called singular if its determinant is:

a) Positive   b) Negative   c) Zero   d) One

Answer: c) Zero

3. The adjoint of a matrix is the transpose of its:

a) Minor matrix   b) Cofactor matrix   c) Inverse   d) Original matrix

Answer: b) Cofactor matrix

4. A square matrix has an inverse only if:

a) det(A) = 0   b) det(A) ≠ 0   c) It is a row matrix   d) It is symmetric

Answer: b) det(A) ≠ 0

5. The formula for the inverse of matrix A is:

a) A⁻¹ = adj(A) × det(A)   b) A⁻¹ = (1/det(A)) × adj(A)   c) A⁻¹ = det(A)/adj(A)   d) A⁻¹ = adj(A) − det(A)

Answer: b) A⁻¹ = (1/det(A)) × adj(A)

Important Short Questions

• Define the determinant of a 2×2 matrix.
• What is meant by a singular matrix?
• Define the minor and cofactor of an element of a matrix.
• State the formula for finding the inverse of a 2×2 matrix.
• Find the determinant of [[6, 2], [3, 1]], and state whether the matrix is singular.

Important Long Questions

• Find the determinant, adjoint, and inverse of A = [[4, 3], [2, 1]], showing all steps.
• Explain the difference between the minor and cofactor of an element, using a 3×3 matrix as an example.
• Determine whether A = [[1, 2], [2, 4]] is singular or non-singular, and explain what this means for its inverse.
• Find the inverse of A = [[5, 2], [3, 1]] and verify your answer by showing A × A⁻¹ equals the identity matrix.

How to Approach This Exercise Effectively

1. Always compute the determinant first — if it’s zero, you can immediately state the matrix has no inverse and stop there.
2. For 2×2 inverses, memorize the adjoint shortcut: swap the diagonal entries and negate the off-diagonal entries.
3. Verify any inverse you compute by multiplying A × A⁻¹ and confirming the result is the identity matrix.
4. Keep minors and cofactors straight by remembering the cofactor only differs from the minor by a sign, based on the checkerboard pattern of (−1)^(i+j).
5. Practice both singular and non-singular examples so you’re comfortable recognizing when a matrix has no inverse.

FAQs

Q: Why do some matrices not have an inverse?

A: A matrix has no inverse whenever its determinant equals zero, since the inverse formula requires dividing by the determinant.

Q: What is the relationship between the cofactor matrix and the adjoint?

A: The adjoint is simply the transpose of the cofactor matrix — every cofactor is calculated first, then the resulting matrix is flipped across its diagonal.

Q: Can a non-square matrix have a determinant?

A: No — determinants are only defined for square matrices, since the concept relies on having an equal number of rows and columns.

Q: What does it mean practically if det(A) = 0?

A: It means the matrix is singular: it has no inverse, and any system of equations represented by that matrix will not have a unique solution.

Notes prepared for Punjab Board (PECTAA) 1st Year Mathematics, 2026-27 session, Chapter 4: Matrices and Determinants, Exercise 4.2.