Class 10 Maths Chapter 3 Exercise 3.4 Solutions 2026 – Matrices and Determinants Notes PDF

Unit 3: Matrices and Determinants | Exercise 3.4 | Punjab Board New Syllabus 2026–27

Updated August 2026: Exercise 3.4 solutions for Matrices and Determinants have been uploaded below and are free to view or download, fully solved according to the new PCTB/PECTAA Class 10 Mathematics syllabus.

Turning a Matrix into a Single Number

Every square matrix has a hidden number attached to it, called its determinant, that summarizes something important about the matrix as a whole — including whether it can be inverted at all. Exercise 3.4 introduces the determinant for the first time, focusing on 2×2 matrices, where the calculation is short enough to do by hand every time.

Key Concepts Covered in Exercise 3.4

What a Determinant Is

The determinant of a matrix A is written |A| or det(A), and unlike the matrix itself, it is always a single real number rather than a grid of numbers.

Determinant of a 2×2 Matrix

For A = [[a, b], [c, d]], the determinant is |A| = ad − bc — multiply the main diagonal entries, then subtract the product of the other diagonal’s entries.

Singular vs. Non-Singular Matrices

A matrix whose determinant equals zero is called singular, and it cannot be inverted. A matrix with a non-zero determinant is called non-singular, and does have an inverse — a distinction that becomes essential in the next exercise.

Properties of Determinants

This exercise introduces a few useful properties: if any row or column is entirely zero, the determinant is zero; and the determinant of the identity matrix is always 1, regardless of its size.

Step-by-Step Solved Examples

Q. No. 1: Find the determinant of A = [[4, 3], [2, 5]].

|A| = ad − bc = (4×5) − (3×2)

|A| = 20 − 6 = 14

Q. No. 2: Determine whether B = [[2, 4], [1, 2]] is singular or non-singular.

|B| = (2×2) − (4×1) = 4 − 4 = 0

Since |B| = 0, matrix B is singular (it has no inverse).

Q. No. 3: Find the value of k for which the matrix [[k, 3], [2, k]] is singular.

For a singular matrix, the determinant must equal 0:

(k×k) − (3×2) = 0

k² − 6 = 0

k² = 6

k = √6  or  k = −√6

These three questions are representative samples. The full PDF notes uploaded on this page walk through every question of Exercise 3.4 in the same numbered, step-by-step format.

MCQs, Short Questions & Long Questions from This Exercise

Alongside the main exercise, this page’s uploaded notes are organized the way Punjab Board papers are structured, so students can practice by question type:

• MCQs: Quick questions calculating the determinant of a small given matrix
• Short Questions: Determining whether a matrix is singular or non-singular
• Long Questions: Finding an unknown value that makes a given matrix singular, like Q. No. 3 above

Where This Matters Beyond This Exercise

The determinant calculated here is exactly what’s needed to find a matrix’s inverse in the next exercise — in fact, a matrix without an inverse is precisely a matrix with a zero determinant. Determinants are also used to check whether a system of two linear equations has a unique solution, appearing later in this chapter’s application to solving equations.

Common Mistakes Students Make in Exercise 3.4

• Subtracting in the wrong order: bc − ad instead of ad − bc
• Confusing the determinant (a number) with the matrix itself (a grid)
• Assuming only a zero matrix can be singular, when many non-zero matrices are singular too
• Sign errors when the matrix contains negative entries

Why This Exercise Matters for the Board Exam

Calculating a 2×2 determinant takes only seconds once ad − bc is memorized, making Exercise 3.4 one of the fastest, most reliable sources of marks in the whole chapter. Singular/non-singular questions are also a favorite short-question style, since they only require one calculation followed by a simple conclusion.

Quick Links – Chapter 3: Matrices and Determinants

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Download Exercise 3.4 Notes PDF

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Frequently Asked Questions (FAQs)

Q1. Are these Exercise 3.4 notes free to download?

Yes, all notes on this page are completely free to view and download in PDF format.

Q2. Which board are these notes for?

These notes are prepared according to the Punjab Board syllabus and are useful for all Punjab boards (Lahore, Gujranwala, Multan, Sargodha, Rawalpindi, Faisalabad, DG Khan, Bahawalpur, Sahiwal) as well as the Federal Board (FBISE).

Q3. What does it mean for a matrix to be “singular”?

A singular matrix is a square matrix whose determinant equals zero. Singular matrices do not have an inverse, which becomes important in the next exercise.

Q4. Does this page include MCQs and short questions too?

Yes, the uploaded notes include MCQs, short questions, and long questions related to this exercise’s topics, organized by question type to match the board paper pattern.

Q5. How can I download the PDF?

Click the “Download PDF” button above and the notes will open or download directly to your device.

Q6. Are these notes updated for the new 2026–27 syllabus?

Yes, these notes are prepared strictly according to the latest PCTB/PECTAA syllabus for the 2026–27 academic session.

Comments & Feedback

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