1. 2.5 Determinants and Multiplicative Inverses of Matrices. Objectives: 1.Evaluate determinants. 2.Find the inverses of matrices. 3.Solve systems of equations by using inverses of matrices.

2. 2.5 Determinants and Multiplicative Inverses of Matrices. Every square matrix has a determinant. The determinant is a value. When the determinant is zero the matrix does not have an inverse. When the determinant is nonzero the matrix has an inverse. To find the determinant of a 2 x 2 matrix: The value of det or Example: Find

3. 2.5 Determinants and Multiplicative Inverses of Matrices. To find the determinant of a 3 x 3 matrix: This is only one minor for a 3 x 3 matrix..

4. 2.5 Determinants and Multiplicative Inverses of Matrices. To find the determinant of a 3 x 3 matrix: Example: Find

5. 2.5 Determinants and Multiplicative Inverses of Matrices. The identity matrix for multiplication of nth order is the square matrix whose elements in the main diagonal, from upper left to lower right are 1s, while all other elements are 0s. The identity matrix for a 2 x 2 is: The identity matrix for a 3 x 3 is:

6. 2.5 Determinants and Multiplicative Inverses of Matrices. The inverse matrix is denoted as First, understand that not every square matrix have an inverse. The product of the a square matrix and its inverse is the identity matrix.

7. 2.5 Determinants and Multiplicative Inverses of Matrices. How do we find the inverse of a matrix? If the determinant is zero then the inverse does not exist. Find the inverse of the matrix, if it exists. Then check to verify it is the inverse. If A = and, then

8. 2.5 Determinants and Multiplicative Inverses of Matrices. Why do we bother with the inverse of a matrix? You have learned how to add, subtract, and multiple matrices together. What if you need to divide matrices? This comes up when we try to solve a system of equations using matrices. If A, B, and X are matrices and AX = B, then how do we isolate the matrix X to find the solution. We can not divide by matrix A, but we can multiple both sides by the inverse of matrix A.

9. 2.5 Determinants and Multiplicative Inverses of Matrices. Let’s solve the system of equations. 4x – y = 1 x + 2y = 7 First rewrite using matrices:

10. 2.5 Determinants and Multiplicative Inverses of Matrices. Find the inverse of matrix A.

11. 2.5 Determinants and Multiplicative Inverses of Matrices. Then multiple both sides by the inverse of matrix A. Therefore, the solution is x = 1 and y = 3 or (1, 3).

12. Date Assignment number 102-104 15-41odd, 42, 44, 49 17 Read pages 106 for tomorrow.