Class 10 Maths Chapter 3 Exercise 3.3 Solutions 2026 – Matrices and Determinants Notes PDF

Unit 3: Matrices and Determinants | Exercise 3.3 | Punjab Board New Syllabus 2026–27

Updated August 2026: Exercise 3.3 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.

A Different Kind of Multiplication

Every matrix operation so far has worked one entry at a time, matching positions directly. Multiplying two matrices together breaks that pattern completely: each entry in the result comes from combining an entire row of the first matrix with an entire column of the second. Exercise 3.3 introduces this new rule, along with the one condition that decides whether two matrices can even be multiplied in the first place.

Key Concepts Covered in Exercise 3.3

The Compatibility Rule

Two matrices A and B can be multiplied (as AB) only if the number of columns in A equals the number of rows in B. If A is m×n and B is n×p, the product AB exists and has order m×p — note that the “inner” numbers must match, and the “outer” numbers become the new order.

Row-by-Column Multiplication

Each entry of the product matrix is found by multiplying corresponding entries of a row from A and a column from B, then adding those products together. The entry in row i, column j of AB comes from row i of A and column j of B.

Matrix Multiplication Is Not Commutative

Unlike ordinary number multiplication, AB does not usually equal BA for matrices — sometimes BA isn’t even defined when AB is. This exercise gives practice recognizing that order matters.

Multiplying by the Identity Matrix

Multiplying any matrix by the appropriately-sized identity matrix leaves it unchanged: AI = IA = A. This mirrors how multiplying an ordinary number by 1 doesn’t change it.

Step-by-Step Solved Examples

Q. No. 1: If A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]], find AB.

Both are 2×2, so AB is defined and will be 2×2.

Position (1,1): (1×5)+(2×7) = 5+14 = 19

Position (1,2): (1×6)+(2×8) = 6+16 = 22

Position (2,1): (3×5)+(4×7) = 15+28 = 43

Position (2,2): (3×6)+(4×8) = 18+32 = 50

AB = [[19, 22], [43, 50]]

Q. No. 2: If A is 2×3 and B is 3×2, what is the order of AB? Is BA defined?

For AB: columns of A (3) = rows of B (3) ✔ defined

Order of AB = rows of A × columns of B = 2×2

For BA: columns of B (2) = rows of A (2) ✔ also defined, order 3×3

Note: AB and BA are defined but have different orders — they cannot be equal.

Q. No. 3: If A = [[2, 0], [1, 3]], verify that AI = A, where I is the 2×2 identity matrix.

I = [[1, 0], [0, 1]]

AI: Position (1,1): (2×1)+(0×0)=2   Position (1,2): (2×0)+(0×1)=0

Position (2,1): (1×1)+(3×0)=1   Position (2,2): (1×0)+(3×1)=3

AI = [[2, 0], [1, 3]] = A ✔ verified

These three questions are representative samples. The full PDF notes uploaded on this page walk through every question of Exercise 3.3 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 on whether two given matrices can be multiplied and what order the result will have
• Short Questions: Multiplying two small square matrices, like Q. No. 1 above
• Long Questions: Verifying properties like AI = A, or showing AB ≠ BA for given matrices

Where This Matters Beyond This Exercise

Matrix multiplication is the foundation for solving systems of linear equations using matrices, and for computing determinants and inverses later in this chapter. In computer science, this exact row-by-column process is how 3D graphics engines rotate, scale, and move objects on screen, and how neural networks process data in machine learning.

Common Mistakes Students Make in Exercise 3.3

• Trying to multiply two matrices without first checking that columns of A equal rows of B
• Multiplying entry-by-entry (like addition) instead of using the row-by-column method
• Assuming AB = BA, which is false for matrices in general
• Losing track of which row and column are being combined partway through a larger multiplication

Why This Exercise Matters for the Board Exam

Matrix multiplication is one of the most heavily tested skills in this chapter, since it’s required for nearly every later topic, including determinants and inverses. Because the compatibility check is quick, examiners often ask students to state whether a product is even defined before asking them to compute it — a fast way to test true understanding rather than mechanical calculation.

Quick Links – Chapter 3: Matrices and Determinants

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Frequently Asked Questions (FAQs)

Q1. Are these Exercise 3.3 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. Why can’t every pair of matrices be multiplied?

Matrix multiplication requires the number of columns in the first matrix to equal the number of rows in the second. If this condition isn’t met, the product simply isn’t defined, no matter what the individual entries are.

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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