Exercise 4.2 Class 11 Math Solutions - Unit 4 Matrices and Determinants, Punjab Board

Step-by-step solutions to Exercise 4.2, Unit 4: Matrices and Determinants, for Class 11 Punjab Board students. Covers calculating minors and cofactors, expanding 3x3 determinants, applying determinant properties and row/column operations, evaluating adjoints, and finding multiplicative inverses of square matrices worked out question by question.

This page provides complete, step-by-step solved solutions for Exercise 4.2 of Unit 4: Matrices and Determinants for Class 11 Mathematics under the Punjab Board (PCTB / PECTAA). All questions involving minors, cofactors, expanding 3x3 determinants along rows and columns, verifying determinant properties (invariance under transpose, row/column interchange sign flipping, scalar factoring, and row/column additions), computing the adjoint matrix adj(A), and evaluating the multiplicative inverse A⁻¹ are thoroughly solved for HSSC Part 1 exam preparation.
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Exercise Overview

Exercise 4.2 deepens the study of Unit 4: Matrices and Determinants by focusing on the numerical and algebraic theory of determinants for square matrices. It starts with calculating the minor Mᵢⱼ and signed cofactor Aᵢⱼ = (-1)ᶦ⁺ʲ Mᵢⱼ for elements of 2x2 and 3x3 matrices, providing the structural basis for Laplace cofactor expansion along any row or column.

A major highlight of this exercise is the rigorous application of determinant properties to simplify evaluation and execute algebraic proofs without direct expansion. Students learn how determinants behave under row and column operations (interchanging rows flips sign, multiplying a row by a scalar scales the determinant, and adding a scalar multiple of one row to another leaves the determinant unchanged). Additionally, the exercise covers constructing the adjoint matrix (the transpose of the cofactor matrix) and computing the multiplicative inverse A⁻¹ = (1/|A|) adj(A) for non-singular matrices.

In Punjab Board HSSC Part 1 examinations, Exercise 4.2 is one of the most critical exercises, consistently featured in Section B short questions (cofactor calculations and 2x2/3x3 inverses) and Section C long questions (proving determinant identities using properties). These step-by-step solutions provide students with the exact algebraic rigor required for top board examination marks.

FAQs

Are these Exercise 4.2 solutions according to the Punjab Board syllabus?
Yes, these solutions strictly adhere to the official Punjab Curriculum and Textbook Board (PCTB / PECTAA) syllabus for Class 11 Mathematics, covering minors, cofactors, determinant expansions, and matrix inverses for the 2026-2027 academic session.
Do these solutions cover every question in Exercise 4.2?
Yes, every problem in Exercise 4.2 is fully solved in sequential order, detailing the computation of 2x2 and 3x3 determinants, cofactor expansions, property-based proofs without direct expansion, and multiplicative inverse calculations.
What is the difference between a minor and a cofactor of a matrix entry?
The minor Mᵢⱼ of an element aᵢⱼ is the determinant of the submatrix obtained by deleting the i-th row and j-th column, while the cofactor is signed: Aᵢⱼ = (-1)ᶦ⁺ʲ Mᵢⱼ.
When does a square matrix have a multiplicative inverse?
A square matrix A possesses a multiplicative inverse A⁻¹ if and only if it is non-singular, meaning its determinant |A| is non-zero (|A| ≠ 0), with the inverse given by A⁻¹ = (1 / |A|) adj(A).
Can students read these solutions online?
Yes, students can read all Exercise 4.2 solutions directly online through MaryamNotes.pk on any web browser without needing to download external files or install third-party apps.
Are these solutions useful for Punjab Board exams?
Yes, evaluating determinants using elementary row/column operations and finding the inverse of 3x3 matrices are among the highest-frequency questions in Section B short questions and Section C long questions for HSSC Part 1.
Are these solutions based on the latest PCTB / PECTAA syllabus?
Yes, they are fully aligned with the updated Student Learning Outcomes (SLO) mandated by the Punjab Curriculum and Textbook Board (PECTAA) for the 2026-2027 academic year.
Are these solutions free for students?
Yes, all Class 11 Mathematics Exercise 4.2 solutions on MaryamNotes.pk are available completely free of charge for students and educators across Punjab.