1. Copyright © Cengage Learning. All rights reserved. 8 Matrices and Determinants

2. 8.4 Copyright © Cengage Learning. All rights reserved. THE DETERMINANT OF A SQUARE MATRIX

3. 3 Find the determinants of 2  2 matrices. Find minors and cofactors of square matrices. Find the determinants of square matrices. What You Should Learn

4. 4 The Determinant of a 2  2 Matrix

5. 5 Every square matrix can be associated with a real number called its determinant. Determinants have many uses, and several will be discussed in this and the next section. Historically, the use of determinants arose from special number patterns that occur when systems of linear equations are solved.

6. 6 The Determinant of a 2  2 Matrix For instance, the system a 1 x + b 1 y = c 1 a 2 x + b 2 y = c 2 has a solution and provided that a 1 b 2 – a 2 b 1  0. Note that the denominators of the two fractions are the same.

7. 7 The Determinant of a 2  2 Matrix This denominator is called the determinant of the coefficient matrix of the system. Coefficient Matrix Determinant det(A) = a 1 b 2 – a 2 b 1

8. 8 The Determinant of a 2  2 Matrix The determinant of the matrix A can also be denoted by vertical bars on both sides of the matrix, as indicated in the following definition.

9. 9 The Determinant of a 2  2 Matrix In this text, det(A) and | A | are used interchangeably to represent the determinant of A. Although vertical bars are also used to denote the absolute value of a real number, the context will show which use is intended. A convenient method for remembering the formula for the determinant of a 2  2 matrix is shown in the following diagram. Note that the determinant is the difference of the products of the two diagonals of the matrix.

10. 10 Example 1 – The Determinant of a 2  2 Matrix Find the determinant of each matrix.

11. 11 Example 1 – Solution = 2(2) – 1(–3) = 4 + 3 = 7 = 2(2) – 4(1) = 4 – 4 = 0

12. 12 Example 1 – Solution = 0(4) – 2 = 0 – 3 = –3 cont’d

13. 13 The Determinant of a 2  2 Matrix Notice in Example 1 that the determinant of a matrix can be positive, zero, or negative. The determinant of a matrix of order 1  1 is defined simply as the entry of the matrix. For instance, if A = [–2], then det(A) = –2.

14. 14 Minors and Cofactors

15. 15 Minors and Cofactors To define the determinant of a square matrix of order 3  3 or higher, it is convenient to introduce the concepts of minors and cofactors. In the sign pattern for cofactors at the left, notice that odd positions (where i + j is odd) have negative signs and even positions (where i + j is even) have positive signs.

16. 16 Example 2 – Finding the Minors and Cofactors of a Matrix Find all the minors and cofactors of Solution: To find the minor M 11, delete the first row and first column of A and evaluate the determinant of the resulting matrix.

17. 17 Example 2 – Solution = –1(1) – 0(2) = –1 Similarly, to find M 12, delete the first row and second column. = 3(1) – 4(2) = –5 cont’d

18. 18 Example 2 – Solution Continuing this pattern, you obtain the minors. M 11 = –1 M 12 = –5 M 13 = 4 M 21 = 2 M 22 = –4 M 23 = –8 M 31 = 5 M 32 = –3 M 33 = –6 Now, to find the cofactors, combine these minors with the checkerboard pattern of signs for a 3  3 matrix shown at the upper left. C 11 = –1 C 12 = 5 C 13 = 4 C 21 = –2 C 22 = –4 C 23 = 8 C 31 = 5 C 32 = 3 C 33 = –6 cont’d

19. 19 The Determinant of a Square Matrix

20. 20 The Determinant of a Square Matrix The definition below is called inductive because it uses determinants of matrices of order n – 1 to define determinants of matrices of order n.

21. 21 The Determinant of a Square Matrix Try checking that for a 2  2 matrix this definition of the determinant yields | A | = a 1 b 2 – a 2 b 1, as previously defined.

22. 22 Example 3 – The Determinant of a Matrix of Order 3  3 Find the determinant of Solution: Note that this is the same matrix that was in Example 2. There you found the cofactors of the entries in the first row to be C 11 = –1, C 12 = 5, andC 13 = 4.

23. 23 Example 3 – Solution So, by the definition of a determinant, you have | A | = a 11 C 11 + a 12 C 12 + a 13 C 13 = 0(–1) + 2(5) + 1(4) = 14. First-row expansion cont’d