Class 10 Maths Chapter 3 Exercise 3.5 Solutions 2026 – Matrices and Determinants Notes PDF

Unit 3: Matrices and Determinants | Exercise 3.5 | Punjab Board New Syllabus 2026–27

Updated August 2026: Exercise 3.5 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.

The Matrix Equivalent of Dividing

Ordinary numbers have a reciprocal: multiply 5 by 1/5 and the result is 1. Matrices have something similar, called the inverse, where multiplying a matrix by its inverse gives the identity matrix. Exercise 3.5 shows how to find this inverse for a 2×2 matrix — building directly on the determinant calculated in Exercise 3.4.

Key Concepts Covered in Exercise 3.5

What an Inverse Matrix Is

The inverse of a square matrix A, written A⁻¹, is the matrix that satisfies AA⁻¹ = A⁻¹A = I, where I is the identity matrix. Only non-singular matrices (those with a non-zero determinant) have an inverse.

The Adjoint of a 2×2 Matrix

For A = [[a, b], [c, d]], the adjoint (written adj A) is formed by swapping the main diagonal entries and changing the sign of the other two: adj A = [[d, −b], [−c, a]].

The Inverse Formula

Once the determinant and adjoint are known, the inverse is A⁻¹ = (1/|A|) × adj A — multiply every entry of the adjoint by the reciprocal of the determinant.

Verifying an Inverse

A good way to check an answer is to multiply A by the calculated A⁻¹ and confirm the result is the identity matrix I. This exercise includes practice doing exactly that.

Step-by-Step Solved Examples

Q. No. 1: Find the inverse of A = [[3, 2], [1, 1]].

First find the determinant: |A| = (3×1) − (2×1) = 3 − 2 = 1

Since |A| ≠ 0, the inverse exists.

adj A = [[1, −2], [−1, 3]]

A⁻¹ = (1/1) × [[1, −2], [−1, 3]] = [[1, −2], [−1, 3]]

Q. No. 2: Find the inverse of B = [[4, 2], [3, 2]].

|B| = (4×2) − (2×3) = 8 − 6 = 2

adj B = [[2, −2], [−3, 4]]

B⁻¹ = (1/2) × [[2, −2], [−3, 4]] = [[1, −1], [−1.5, 2]]

Q. No. 3: Verify that AA⁻¹ = I for A = [[3, 2], [1, 1]] and A⁻¹ = [[1, −2], [−1, 3]] (from Q. No. 1).

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

Position (1,2): (3×−2)+(2×3) = −6+6 = 0

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

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

AA⁻¹ = [[1, 0], [0, 1]] = I ✔ verified

These three questions are representative samples. The full PDF notes uploaded on this page walk through every question of Exercise 3.5 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 the adjoint of a given 2×2 matrix
• Short Questions: Finding the inverse of a matrix with a simple determinant, like Q. No. 1 above
• Long Questions: Finding an inverse and then verifying it by multiplication, like Q. No. 2 and 3 combined

Where This Matters Beyond This Exercise

The matrix inverse is the key tool used in the next exercise to solve systems of linear equations directly, without substitution or elimination. Outside the classroom, matrix inverses are used in cryptography for encoding and decoding messages, and in computer graphics to reverse transformations like rotation or scaling.

Common Mistakes Students Make in Exercise 3.5

• Trying to find an inverse for a matrix whose determinant is zero (singular matrices have no inverse)
• Swapping the wrong diagonal when forming the adjoint, or forgetting to change signs on the off-diagonal
• Forgetting to multiply every entry of the adjoint by 1/|A|
• Making an arithmetic slip when the determinant is a fraction or negative number

Why This Exercise Matters for the Board Exam

Finding a 2×2 inverse is a compact, formulaic long question that combines two full skills — the determinant from Exercise 3.4 and the new adjoint step — making it one of the more heavily weighted question types in this chapter. Examiners often ask students to verify their answer by multiplication, so practicing that final check, as in Q. No. 3, is worth the extra time.

Quick Links – Chapter 3: Matrices and Determinants

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Frequently Asked Questions (FAQs)

Q1. Are these Exercise 3.5 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. Does every matrix have an inverse?

No — only square, non-singular matrices (those with a non-zero determinant) have an inverse. A singular matrix (determinant = 0) has no inverse at all.

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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