Recall that a square matrix is one in which there are the same amount of rows as columns. A square matrix must exist in order to evaluate a determinant. A determinant is a real number that is associated with a square matrix, and is denoted by x. The examples that follow will deal with a 2 x 2 and a 3 x 3 matrix. The determinant of a 2 x 2 matrix is found by using the following definition Definition 1 If A = , then Next Slide Definition 1 (Alternate) If A = , then This notation may seem confusing to some. Here is an alternate notation.
Example 1a. If A = Solution: Example 1b. Evaluate Solution: Note: Finding the determinant is often called evaluating a determinant, and the matrix notation is often omitted. Your Turn Problem #1 Evaluate the determinants a) b) Answer
We will now discuss how to evaluate a 33 determinant We will now discuss how to evaluate a 33 determinant. We will show two separate methods of evaluating these determinants. This first method is a cross multiplication method and the second method is called ‘expansion by minors’. Many find the first method easier. However, if we needed to evaluate a 44 determinant, we would have to use expansion by minors. The directions will be confusing to understand, however the example will clarify the procedure. Also, if you are planning to go on in mathematics, you should learn both methods. Cross Multiplication Method. 1. Rewrite the first two columns to the right of the 33 determinant. Add the following 3 diagonals in the following order: Multiply the diagonal from the 1st number on the top row to the 3rd number on the bottom row. Then multiply the diagonal from the 2nd number on the top row to the 4th number on the bottom row. Next multiply the diagonal from the 3rd number on the top row to the 5th number on the bottom row. Subtract the following 3 diagonals in the following order: (change the sign of the result of the following diagonals.) Multiply the diagonal from the 5th number on the top row to the 3rd number on the bottom row. Then multiply the diagonal from the 4th number on the top row to the 2nd number on the bottom row. Next multiply the diagonal from the 3rd number on the top row to the 1st number on the bottom row. Remember to change the sign of each of these 3 answers. 4. Add the 6 diagonals and this is the value of the determinant.
Example 2. Evaluate: Solution: 1. Rewrite 1st two columns. 2. Add the first 3 diagonals. 12 30 + 12 45 12 8 = 95 Next, subtract the next 3 diagonals. Example 2B. Evaluate: Solution: 1. Rewrite 1st two columns. 2. Add the first 3 diagonals. 30 + 0 + 12 +0 5 +16 = 7 Next, subtract the next 3 diagonals. Your Turn Problem #2 Evaluate the determinants a) b) Answer
Expansion by Minors Method. To learn this method, the first item to discuss is finding a “minor”. A minor is the determinant of a 2 x 2 matrix contained within a larger matrix (in this case, a 3 x 3) that is obtained by selecting an element (or number) within the matrix and “deleting” the row and column in which the element appears. The remaining elements make up the determinate for the minor. Example: Find the minor of 2. Example: Find the minor of 6. 1st, cross out the row and column in which the 2 appears. Then, is the the minor of 2. Cross out the row and column in which the 6 appears. Then, is the the minor of 6.
Next we will use this Array of Signs for a 3 x 3 Determinant These signs will be inserted in front of any element from a chosen row or column. Procedure for Evaluating a 3 x 3 determinant by expansion by minors. 1. Choose any row or column. Write the first element with its corresponding sign from the array from the row or column an multiply it by its minor. Continue with the next two elements of the row or column. 3. Do all necessary calculations to add these products together to obtain the value of the 3 x 3 determinant. Next Slide
Example 3. Evaluate using expansion by minors: Solution: We can choose any row or column along with the array of signs. Let’s use the top row. Using the procedure and the array of signs we obtain: Answer: 95
Example 3. Again by minors: (Let’s do the same example using the first column.) Solution: Using the procedure and the array of signs we obtain: Answer: 95 Your Turn Problem #3 Evaluate the determinants by expansion by minors. a) b) Answer a) 6 b) 17
Cramer’s rule is another tool that may be used to solve a system of linear equations. Solving a 2 x 2 determinants will require three separate determinants to be evaluated. Solving a 3 x 3 determinant will require four separate determinants to be evaluated. Therefore, to complete this section, you must be proficient in evaluating determinants. The previous section contained two methods for evaluating determinants. Please review if necessary. Using Cramer’s Rule to Solve a 2 x 2 System of Equations Using the elimination method and the definition of determinants, we have the following rule: Cramer’s Rule (2 x 2 case) Note: If D= 0 and Dx or Dy does not equal zero, the system is inconsistent. The solution set is . If D, Dx and Dy are all zero then the equations are dependent and there are infinitely many solutions. Next Slide
Solution: Example 4. Solve the system by Cramer’s Rule: The solution set is . Solve the system by Cramer’s Rule: Your Turn Problem #4 Answer:
Cramer’s Rule (3 x 3 case) Using Cramer’s Rule to Solve a 3 x 3 System of Equations Next Slide Note: If D= 0 and Dx, Dy or Dz does not equal zero, the system is inconsistent. The solution set is . If D, Dx, Dy, and Dz are all zero then the equations are dependent and there are infinitely many solutions.
Example 5. Solve the system by Cramer’s Rule: Solution: The solution set is . Solve the system by Cramer’s Rule: Your Turn Problem #5 Then End B.R. 8-09-04 Answer: