Exercise 3.4 Class 10 Math Solutions – Matrix Multiplication & Laws, Punjab Board

Solved Exercise 3.4 for Class 10 Mathematics, Punjab Board. Complete step-by-step notes covering matrix multiplication conformability, row-by-column product calculations, non-commutativity (AB ≠ BA), associative law ((AB)C = A(BC)), distributive laws (A(B ± C) = AB ± AC), and multiplicative identity (AI = IA = A).

This page provides complete solved solutions for Exercise 3.4 from Unit 3: Matrices and Determinants, Class 10 Mathematics, Punjab Board (PCTB). It covers matrix multiplication conditions (columns of A = rows of B), step-by-step row-by-column product computations, proving that matrix multiplication is generally not commutative (AB ≠ BA), verifying the associative law ((AB)C = A(BC)), distributive laws over addition and subtraction (A(B+C)=AB+AC), and the multiplicative identity matrix (I). All questions are worked out step by step for Matric Part 2 students.
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Exercise Overview

Exercise 3.4 is one of the most critical conceptual and computational chapters in Unit 3: Matrices and Determinants for Class 10 Punjab Board Mathematics. It shifts focus from element-wise addition to row-by-column matrix multiplication, introducing students to the non-intuitive algebraic structures of modern linear algebra.

The exercise begins with the compatibility test for multiplication: the product AB exists if and only if the number of columns of matrix A equals the number of rows of matrix B. Students master the row-by-column multiplication process, calculating dot products of vectors to determine entry values. Key theoretical investigations include demonstrating the general non-commutativity of matrix multiplication (AB ≠ BA), proving the Associative Law ((AB)C = A(BC)), applying the Left and Right Distributive Laws (A(B + C) = AB + AC and (A + B)C = AC + BC), and establishing the role of the identity matrix I as the multiplicative identity (AI = IA = A).

In Punjab Board examinations (such as Lahore, Faisalabad, Rawalpindi, Gujranwala, Multan, and other BISE boards), Exercise 3.4 questions regularly appear as high-scoring long questions in Section C. Verifying associative and distributive laws requires meticulous arithmetic and step-by-step documentation. These comprehensive solutions provide clear, structured working to ensure matric students achieve full marks.

FAQs

Are these Exercise 3.4 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 3.4 matrix multiplication topics for the 2026-2027 session.
When are two matrices conformable for multiplication?
Two matrices A and B are conformable for multiplication AB if the number of columns in the first matrix A is equal to the number of rows in the second matrix B.
How do you multiply two matrices in Exercise 3.4?
Matrix multiplication is performed using the row-by-column method: multiply each entry of a row of the first matrix by the corresponding entry of a column of the second matrix, and sum these products.
Is matrix multiplication commutative in general?
No. In general, matrix multiplication is not commutative, meaning AB ≠ BA (except in special cases like identity or inverse matrices).
What is the associative law of matrix multiplication?
For any three conformable matrices A, B, and C, the associative law holds: (AB)C = A(BC).
What are the distributive laws of matrix multiplication?
Matrix multiplication distributes over addition and subtraction: A(B + C) = AB + AC (left distributive law) and (A + B)C = AC + BC (right distributive law).
What is the multiplicative identity of a matrix?
For any square matrix A of order n × n, the identity matrix I of the same order is the multiplicative identity, satisfying AI = IA = A.
Can students read these Exercise 3.4 solutions online?
Yes, the complete solved PDF notes for Exercise 3.4 are embedded to read online directly on MaryamNotes.pk with no compulsory download.
Are these Exercise 3.4 solutions free for students?
Yes, all Exercise 3.4 solutions on MaryamNotes.pk are 100% free to read online for all Class 10 students across all Punjab boards.