Exercise 3.5 Class 10 Math Solutions – Determinants, Adjoint & Inverses, Punjab Board

Solved Exercise 3.5 for Class 10 Mathematics, Punjab Board. Detailed step-by-step notes covering determinants of 2×2 matrices, singular vs non-singular classification, finding the adjoint of a matrix, and computing multiplicative inverses using the adjoint method.

This page provides complete solved solutions for Exercise 3.5 from Unit 3: Matrices and Determinants, Class 10 Mathematics, Punjab Board (PCTB). It covers finding determinants of 2×2 matrices (det(A) = ad - bc), distinguishing singular (det(A) = 0) from non-singular (det(A) ≠ 0) matrices, finding adjoint matrices (adj(A)), and calculating the multiplicative inverse using the adjoint formula (A⁻¹ = adj(A) / det(A)). All questions are worked out step by step for Matric Part 2 students.
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Exercise Overview

Exercise 3.5 represents an essential core topic in Unit 3: Matrices and Determinants for Class 10 Punjab Board Mathematics. It introduces students to determinants and matrix inversion, which serve as the algebraic foundation for solving simultaneous linear equations in subsequent exercises.

The exercise systematically guides students through evaluating determinants of 2×2 matrices using the cross-multiplication formula |A| = ad - bc. Based on determinant values, matrices are classified as singular (|A| = 0) or non-singular (|A| ≠ 0). Students master computing the adjoint of a matrix by transposing cofactor signs and positions, and then apply the Adjoint Method to calculate multiplicative inverses A⁻¹ = adj(A)/|A|. The exercise also reinforces theoretical verification by demonstrating that multiplying a matrix by its inverse yields the multiplicative identity matrix (A A⁻¹ = A⁻¹ A = I).

In Punjab Board examinations (including Lahore, Faisalabad, Rawalpindi, Gujranwala, Multan, and other BISE boards), Exercise 3.5 problems are frequently tested in Section B short questions (calculating determinants and adjoints) and Section C long questions (finding inverses and verifying inverse identities). These step-by-step solutions provide precise mathematical structure to help students achieve top scores in their matric examinations.

FAQs

Are these Exercise 3.5 solutions according to the Punjab Board syllabus?
Yes. These solutions follow the Punjab Curriculum and Textbook Board (PCTB) syllabus for Class 10 Mathematics, covering all Unit 3 Exercise 3.5 determinants and inverse topics for the 2026-2027 session.
How do you calculate the determinant of a 2×2 matrix?
For a matrix A = [[a, b], [c, d]], the determinant is calculated as |A| = det(A) = (a)(d) - (b)(c) = ad - bc.
What is the difference between a singular and a non-singular matrix?
A square matrix is singular if its determinant is zero (|A| = 0), and non-singular if its determinant is non-zero (|A| ≠ 0).
Why is the inverse of a singular matrix not possible?
Because the formula for the inverse requires dividing by the determinant (A⁻¹ = adj(A)/|A|), and division by zero is undefined, a singular matrix has no multiplicative inverse.
How do you find the adjoint of a 2×2 matrix in Exercise 3.5?
For a matrix A = [[a, b], [c, d]], the adjoint adj(A) is found by interchanging the diagonal entries (a and d) and changing the signs of the non-diagonal entries (b and c), giving [[d, -b], [-c, a]].
What is the formula to find the multiplicative inverse of a matrix?
The multiplicative inverse of a non-singular square matrix A is given by A⁻¹ = (1 / |A|) × adj(A).
How do you verify if matrix B is the multiplicative inverse of matrix A?
Compute the products AB and BA; if AB = BA = I (where I is the identity matrix of the same order), then B is the multiplicative inverse of A.
Can students read these Exercise 3.5 solutions online?
Yes, the complete solved PDF notes for Exercise 3.5 are embedded to read online directly on MaryamNotes.pk with no compulsory download.
Are these Exercise 3.5 solutions free for students?
Yes, all Exercise 3.5 solutions on MaryamNotes.pk are 100% free to read online for all Class 10 students across all Punjab boards.