Determinants

Determinants assign a single number to every square matrix, and this number decides whether the matrix has an inverse and whether a system of linear equations has a unique solution. In this chapter you will learn to evaluate determinants of order 1, 2 and 3 by expanding along any row or column, and to use determinants to find the area of a triangle whose vertices are given, to test whether three points are collinear, and to find the equation of a line through two points. You will then study minors and cofactors, the adjoint of a matrix, and the key result A(adj A) = (adj A)A = det(A) I. From this you will find the inverse of a non-singular matrix as A^-1 = (1/det A) adj A. The chapter ends with solving a system of linear equations in two or three variables by the matrix method, X = A^-1 B, and testing consistency: a unique solution when det A is non-zero, and no solution or infinitely many solutions when det A = 0.

Detailed Worksheet: Determinants Section A - Definitions (10 marks) 1. Define the minor and the cofactor of an element of a determinant. Find the minor and cofactor of the element 4 in the matrix [[1, 2], [4, 3]]. (2 marks) 2. What is a singular matrix? Is the matrix [[2, 4], [1, 2]] singular? (2 marks) 3. Define the adjoint of a square matrix. Write the adjoint of [[1, 2], [3, 4]]. (2 marks) 4. If A is a square matrix of order 3 and det A = 5, find det(2A) and det(adj A). (2 marks) 5. State the condition for a system of linear equations AX = B to have a unique solution. What happens when det A = 0 and (adj A)B is not the zero matrix? (2 marks) Section B - Calculations and Applications (15 marks) 6. Evaluate the determinant of the matrix [[1, 2, 3], [0, 4, 5], [1, 0, 6]] by expanding along the first column. (3 marks) 7. Using determinants, find the area of the triangle with vertices (1, 0), (6, 0) and (4, 3). (3 marks) 8. Find the values of k if the area of the triangle with vertices (k, 0), (4, 0) and (0, 2) is 4 square units. (3 marks) 9. Find the inverse of the matrix A = [[2, -2], [4, 3]] and verify that A A^-1 = I. (3 marks) 10. Solve the system 2x + 5y = 1 and 3x + 2y = 7 by the matrix method. (3 marks) Section C - Diagrams (10 marks) 11. Plot the points (1, 0), (6, 0) and (4, 3) on a graph, draw the triangle, and verify the area found in question 7 using the formula one-half times base times height. (4 marks) 12. Draw a flowchart showing the steps to find the inverse of a 3 x 3 matrix: find det A, check non-singularity, find cofactors, form adj A, and divide by det A. (3 marks) 13. Draw a chart showing the three cases for the consistency of AX = B: det A non-zero (unique solution), det A = 0 with (adj A)B non-zero (no solution), and det A = 0 with (adj A)B = O (infinitely many or no solution, to be checked further). (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Solve the system x - y + 2z = 7, 3x + 4y - 5z = -5 and 2x - y + 3z = 12 using the matrix method. (5 marks) 15. For A = [[1, -1, 2], [3, 0, -2], [1, 0, 3]], find adj A and verify that A(adj A) = (adj A)A = det(A) I. Hence write A^-1. (5 marks) 16. Examine the consistency of the following systems and explain your reasoning using determinants: (i) x + 3y = 5 and 2x + 6y = 8, (ii) x + y = 2 and 2x + 2y = 4, (iii) 2x - y = 5 and x + y = 4. Interpret each system geometrically as a pair of lines. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Show all cofactors and steps clearly. Verify solutions by substitution where possible.