1 MAC 2103 Module 3 Determinants

2 Rev.F09 Learning Objectives Upon completing this module, you should be able to: 1. Determine the minor, cofactor, and adjoint of a matrix. 2. Evaluate the determinant of a matrix by cofactor expansion. 3. Determine the inverse of a matrix using the adjoint. 4. Solve a linear system using Cramer’s Rule. 5. Use row reduction to evaluate a determinant. 6. Use determinants to test for invertibility. 7. Find the eigenvalues and eigenvectors of a matrix. Click link to download other modules.

3 Rev.09 Determinants Click link to download other modules. Determinants by Cofactor Expansion Evaluating Determinants by Row Reduction Properties of the Determinant There are three major topics in this module:

4 Rev.F09 What is a Determinant? Click link to download other modules. Determinants are commonly used to test if a matrix is invertible and to find the area of certain geometric figures. A determinant is a real number associated with a square matrix.

5 Rev.F09 How to Determine if a Matrix is Invertible? Click link to download other modules. The following is often used to determine if a square matrix is invertible.

6 Rev.F09 Example Click link to download other modules. Determine if A -1 exists by computing the determinant of the matrix A. a)b) Solution a) b) A -1 does not exist A -1 does exist

7 Rev.F09 What are Minors and Cofactors? Click link to download other modules. We know we can find the determinants of 2 x 2 matrices; but can we find the determinants of 3 x 3 matrices, 4 x 4 matrices, 5 x 5 matrices,...? In order to find the determinants of larger square matrices, we need to understand the concept of minors and cofactors.

8 Rev.F09 Example of Finding Minors and Cofactors Click link to download other modules. Find the minor M 11 and cofactor A 11 for matrix A. Solution To obtain M 11 begin by crossing out the first row and column of A. The minor is equal to det B = −6(5) − (−3)(7) = −9 Since A 11 = (−1) 1+1 M 11, A 11 can be computed as follows: A 11 = (−1) 2 (−9) = −9

9 Rev.F09 How to Find the Determinant of Any Square Matrix? Click link to download other modules. Once we know how to obtain a cofactor, we can find the determinant of any square matrix. You may pick any row or column, but the calculation is easier if some elements in the selected row or column equal 0. for any column jfor any row i or

10 Rev.F09 Example of Finding the Determinant by Cofactor Expansion Click link to download other modules. Find det A, if Solution To find the determinant of A, we can select any row or column. If we begin expanding about the first column of A, then det A = a 11 A 11 + a 21 A 21 + a 31 A 31. A 11 = −9 from the previous example A 21 = −12 and A 31 = 24 det A = a 11 A 11 + a 21 A 21 + a 31 A 31 = (−8)(−9) + (4)(−12) + (2)(24) = 72 Now, try to find the determinant of A by expanding the first row of A.

11 Rev.F09 How to Find the Adjoint of a Matrix? Click link to download other modules. The adjoint of a matrix can be found by taking the transpose of the matrix of cofactors from A. In our previous example, we have found the cofactors A 11, A 21, A 31. If we continue to solve for the rest of the cofactors for matrix A, namely A 12, A 22, A 32, A 13, A 23, and A 33, then we will have a 3 x 3 matrix of cofactors from A as follows:

12 Rev.F09 How to Find the Adjoint of a Matrix? (Cont.) Click link to download other modules. The transpose of this 3 x 3 matrix of cofactors from A is called the adjoint of A, and it is denoted by Adj(A). What are we going to do with this Adj(A)? We can use it to help us find the A -1 if A is an invertible matrix.

13 Rev.F09 How to Find A -1 Using the Adjoint of a Matrix? Click link to download other modules. Theorem 2.1.2: If A is an invertible matrix, then Note: 1.The square matrix A is invertible if and only if det(A) is not zero. 2. If A is an n x n triangular matrix, then det(A) is the product of the entries on the main diagonal of the matrix (Theorem )

14 Rev.F09 What is Cramer’s Rule? Click link to download other modules. Cramer’s Rule is a method that utilizes determinants to solve systems of linear equations. This rule can be extended to a system of n linear equations in n unknowns as long as the determinant of the matrix is non-zero.

15 Rev.F09 Example of Using Cramer’s Rule to Solve the Linear System Click link to download other modules. Use Cramer’s rule to solve the linear system. Solution In this system a 1 = 1, b 1 = 4, c 1 = 3, a 2 = 2, b 2 = 9 and c 2 = 5

16 Rev.F09 Example of Using Cramer’s Rule to Solve the Linear System (cont.) Click link to download other modules. E = 7, F = −1 and D = 1 The solution is Note that Gaussian elimination with backward substitution is usually more efficient than Cramer’s Rule.

17 Rev.F09 What Are the Limitations on the Method of Cofactors and Cramer’s Rule? The main limitations are as follow: 1. A substantial number of arithmetic operations are needed to compute determinants of large matrices. 2. The cofactor method of calculating the determinant of an n x n matrix, n > 2, generally involves more than n! multiplication operations. 3. Time and cost required to solve linear systems that involve thousands of equations in real-life applications. Next, we are going to look at a more efficient method to find the determinant of a general square matrix. Click link to download other modules.

18 Rev.F09 Evaluating Determinants by Reducing the Matrix to Row-Echelon Form Click link to download other modules. Let A be a square matrix. (See Theorem 2.2.3) (a) If B is the matrix that results from scaling by a scalar k, then det(B) = k det(A). (b) If B is the matrix that results from either rows interchange or columns interchange, then det(B) = - det(A). (c) If B is the matrix that results from row replacement, then det(B) = det(A). Just keep these in mind when A is a square matrix: 1. det(A)=det(A T ). 2. If A has a row of zeros or a column of zeros, then det(A)=0. 3. If A has two proportional rows or two proportional columns, then det(A)=0.

19 Rev.F09 How to Evaluate the Determinant by Row Reduction? Click link to download other modules. Let’s look at a square matrix A. We can find the determinant by reducing it into row-echelon form. Step 1: We want a leading 1 in row 1. We can interchange row 1 and row 2 to accomplish this.

20 Rev.F09 How to Evaluate the Determinant by Row Reduction? (Cont.) Click link to download other modules. Step 2: We want a leading 1 in row 2. We can take a common factor of 3 from row 2 to accomplish this (scaling). Step 3: We want a zero at both row 2 and row 3 below the leading 1 in row 1. We can add -3 times row 1 to row 3 to accomplish this (row replacement). From Step 1:

21 Rev.F09 How to Evaluate the Determinant by Row Reduction? (Cont.) Click link to download other modules. Step 4: We want a zero below the leading 1 in row 2. We can add row 2 to row 3 to accomplish this (row replacement). Step 5: We want a leading 1 in row 3. We take a common factor of -5/3 from row 3 to accomplish this (scaling). From Step 3: Remember: If A is an n x n triangular matrix, then det(A) is the product of the entries on the main diagonal of the matrix.

22 Rev.F09 Let’s Look at Some Useful Basic Properties of Determinants Click link to download other modules. Let A and B be n x n matrices and k is any scalar. Then, If A is invertible, then This is because A -1 A=I, det(A -1 A) =det(I) =1; det(A -1 ) det(A) = 1, and so Question: Is det(A+B) = det(A) + det(B) ? Remember: If A is an n x n triangular matrix, then det(A) is the product of the entries on the main diagonal of the matrix.

23 Rev.F09 What are Eigenvalues and EigenVectors? Click link to download other modules. An eigenvector of an n x n matrix A is a nontrivial (nonzero) vector such that, where is a scalar called an eigenvalue. Linear systems of this form can be rewritten as follows: The system has a nontrivial solution if and only if This is the so called characteristic equation of A and therefore B has no inverse, and the linear system has infinitely many solutions.

24 Rev.F09 Click link to download other modules. Express the following linear system in the form Find the characteristic equation, eigenvalues and eigenvectors corresponding to each of the eigenvalues. The linear system can be written in matrix form as with Example

25 Rev.F09 Click link to download other modules. which is of the form Thus, Can you tell what is the characteristic equation for A? Example (Cont.)

26 Rev.F09 Click link to download other modules. The characteristic equation for A is or Example (Cont.)

27 Rev.F09 Click link to download other modules. Thus, the eigenvalues of A are: By definition, is an eigenvector of A if and only if is a nontrivial solution of that is If, then we have Thus, we can form the augmented matrix and solve by Gauss Jordan Elimination. Example (Cont.)

28 Rev.F09 Click link to download other modules. Let’s form the augmented matrix and solve by Gauss Jordan Elimination. Thus, a free variable, Example (Cont.)

29 Rev.F09 Click link to download other modules. Solving this system yields: So the eigenvectors corresponding to are the nontrivial solutions of the form Similarly, if, then we have Example (Cont.)

30 Rev.F09 Click link to download other modules. Example (Cont.) Let’s form the augmented matrix and solve by Gauss Jordan Elimination. Thus,

31 Rev.F09 Click link to download other modules. Solving this system yields: So the eigenvectors corresponding to are the nontrivial solutions of the form Example (Cont.)

32 Rev.F09 What have we learned? We have learned to: 1. Determine the minor, cofactor, and adjoint of a matrix. 2. Evaluate the determinant of a matrix by cofactor expansion. 3. Determine the inverse of a matrix using the adjoint. 4. Solve a linear system using Cramer’s Rule. 5. Use row reduction to evaluate a determinant. 6. Use determinants to test for invertibility. 7. Find the eigenvalues and eigenvectors of a matrix. Click link to download other modules.

33 Rev.F09 Credit Some of these slides have been adapted/modified in part/whole from the text or slides of the following textbooks: Anton, Howard: Elementary Linear Algebra with Applications, 9th Edition Rockswold, Gary: Precalculus with Modeling and Visualization, 3th Edition Click link to download other modules.