Matrices

A matrix is an ordered rectangular array of numbers, and matrices give a compact way to store data and to handle many linear equations at once, with uses in economics and cryptography. In this chapter you will learn the order of a matrix, how to construct a matrix from a rule for its elements, and the types of matrices: row, column, square, diagonal, scalar, identity and zero matrices. You will learn when two matrices are equal and use this to find unknown entries. You will then study operations on matrices: addition, scalar multiplication and multiplication, with their properties, and see that matrix multiplication is not commutative and that AB = O does not imply A = O or B = O. The chapter covers the transpose of a matrix and its properties, including (AB)^T = B^T A^T, symmetric and skew symmetric matrices, and the result that every square matrix can be written as the sum of a symmetric and a skew symmetric matrix. Finally, you will learn about invertible matrices and the uniqueness of the inverse.

Detailed Worksheet: Matrices Section A - Definitions (10 marks) 1. Define a square matrix and a diagonal matrix with one example each. (2 marks) 2. What is a scalar matrix? How is it different from an identity matrix? (2 marks) 3. Define a symmetric matrix and a skew symmetric matrix. Why are the diagonal elements of a skew symmetric matrix zero? (2 marks) 4. If A is of order 3 x 4 and B is of order 4 x 2, what is the order of AB? Is BA defined? (2 marks) 5. Define an invertible matrix. Prove that the inverse of a square matrix, if it exists, is unique. (2 marks) Section B - Calculations and Applications (15 marks) 6. If A = [[1, 2], [3, 4]] and B = [[2, 0], [1, 3]], find A + B and 2A - B. (3 marks) 7. For the matrices in question 6, find AB and BA. Is AB = BA? (3 marks) 8. Find x and y if [[x + y, 2], [5, x - y]] = [[6, 2], [5, 2]]. (3 marks) 9. Express A = [[3, 5], [1, -1]] as the sum of a symmetric and a skew symmetric matrix. (3 marks) 10. If A = [[3, 1], [-1, 2]], show that A^2 - 5A + 7I = O. (3 marks) Section C - Diagrams (10 marks) 11. Draw a chart classifying matrices into row, column, square, diagonal, scalar, identity and zero matrices with one example of each. (4 marks) 12. Draw a diagram showing how the element in row 2, column 1 of AB is obtained from row 2 of A and column 1 of B for A = [[1, 2], [3, 4]] and B = [[2, 0], [1, 3]]. (3 marks) 13. Draw a flowchart to decide whether a given square matrix is symmetric, skew symmetric or neither, using its transpose. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Construct a 2 x 3 matrix whose elements are given by a_ij = (i + 2j)^2/2. Find its transpose and state whether the product of the matrix with its transpose is symmetric. (5 marks) 15. If A = [[1], [-4], [3]] and B = [[-1, 2, 1]], verify that (AB)^T = B^T A^T. (5 marks) 16. Two shops sell pens, notebooks and files. Shop 1 sells 10 pens, 2 notebooks and 5 files; shop 2 sells 5 pens, 10 notebooks and 8 files. The selling prices are Rs 10, Rs 40 and Rs 25, and the cost prices are Rs 8, Rs 30 and Rs 20 respectively. Using matrix multiplication, find the revenue, cost and profit of each shop. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Write matrices row by row in square brackets and show each product element clearly.