Class 10 Maths Chapter 3 Exercise 3.6 Solutions 2026 – Matrices and Determinants Notes PDF

Unit 3: Matrices and Determinants | Exercise 3.6 | Punjab Board New Syllabus 2026–27

Updated August 2026: Exercise 3.6 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.

Where All of Chapter 3 Comes Together

A pair of linear equations in two variables can be solved by substitution or elimination, as in earlier classes — but it can also be rewritten entirely as a single matrix equation and solved using everything learned in this chapter. Exercise 3.6 is where matrix order, multiplication, determinants, and inverses all combine into one practical method.

Key Concepts Covered in Exercise 3.6

Writing Equations in Matrix Form

A system like ax + by = e and cx + dy = f can be written as AX = B, where A = [[a, b], [c, d]], X = [[x], [y]], and B = [[e], [f]]. Multiplying A by X using matrix multiplication reproduces exactly the original two equations.

Solving Using the Inverse

Once AX = B is set up, both sides can be multiplied by A⁻¹ (found the way Exercise 3.5 taught): A⁻¹AX = A⁻¹B, which simplifies to X = A⁻¹B, since A⁻¹A = I. Calculating A⁻¹B directly gives the values of x and y.

Checking for a Unique Solution

This method only works when A is non-singular (|A| ≠ 0). If |A| = 0, the matrix has no inverse, meaning the system either has no solution or infinitely many — the same idea introduced back in Exercise 3.4.

Step-by-Step Solved Examples

Q. No. 1: Solve using matrices: 2x + y = 5 and x + 3y = 10.

Write in matrix form: A = [[2, 1], [1, 3]], X = [[x], [y]], B = [[5], [10]]

Find |A| = (2×3) − (1×1) = 6 − 1 = 5 (non-zero, so solution exists)

adj A = [[3, −1], [−1, 2]]

A⁻¹ = (1/5)[[3, −1], [−1, 2]]

X = A⁻¹B = (1/5)[[3, −1], [−1, 2]] × [[5], [10]]

x = (1/5)[(3×5)+(−1×10)] = (1/5)(15−10) = 1

y = (1/5)[(−1×5)+(2×10)] = (1/5)(−5+20) = 3

Solution: x = 1, y = 3

Q. No. 2: Verify the solution to Q. No. 1 by substituting back into the original equations.

Check equation 1: 2(1) + 3 = 2 + 3 = 5 ✔ matches

Check equation 2: 1 + 3(3) = 1 + 9 = 10 ✔ matches

Both equations are satisfied, confirming x = 1, y = 3 is correct.

Q. No. 3: Does the system 4x + 2y = 6 and 2x + y = 3 have a unique solution using this method?

A = [[4, 2], [2, 1]]

|A| = (4×1) − (2×2) = 4 − 4 = 0

Since |A| = 0, A has no inverse, so this method cannot find a unique solution.

(In fact, the second equation is exactly half the first, so infinitely many solutions exist.)

These three questions are representative samples. The full PDF notes uploaded on this page walk through every question of Exercise 3.6 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 writing a given system of equations in the form AX = B
• Short Questions: Checking whether a system has a unique solution using the determinant
• Long Questions: Full matrix-method solutions like Q. No. 1, including finding A⁻¹ and computing X

Where This Matters Beyond This Exercise

This exercise is the practical payoff for the whole chapter, showing why order, multiplication, determinants, and inverses were worth learning in the first place. The same matrix-inverse method scales up to solve much larger systems of equations in engineering, economics, and computer science, where solving by hand would otherwise be impractical.

Common Mistakes Students Make in Exercise 3.6

• Setting up matrix A with rows and columns swapped from the original equations
• Forgetting to check |A| ≠ 0 before assuming a unique solution exists
• Making an arithmetic slip in the final A⁻¹B multiplication
• Skipping the verification step, which catches most calculation errors quickly

Why This Exercise Matters for the Board Exam

Because this exercise draws on every earlier skill in the chapter, it’s one of the most heavily weighted long-question types in Matrices and Determinants. Presenting the solution in clear stages — matrix form, determinant check, inverse, then X = A⁻¹B — tends to earn strong method marks even if a small slip affects the final numbers.

Quick Links – Chapter 3: Matrices and Determinants

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Download Exercise 3.6 Notes PDF

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Frequently Asked Questions (FAQs)

Q1. Are these Exercise 3.6 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. Do I need Exercises 3.4 and 3.5 before starting this one?

Yes — this exercise directly uses the determinant (Exercise 3.4) and inverse (Exercise 3.5) methods to solve equations, so both should be comfortable before starting here.

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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