The Inverse of a Matrix

1. The Inverse of a Matrix Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

2. The inverse of a square matrix A is another matrix with the following properties: Here I represents the identity matrix of the same size as A and A-1. Note that A-1 must be a square matrix of the same size as A. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

3. The inverse of a square matrix A is another matrix with the following properties: Here I represents the identity matrix of the same size as A and A-1. Note that A-1 must be a square matrix of the same size as A. Here is a system of linear equations. To solve it, we can put it into matrix format and try to find the inverse of the coefficient matrix. Let’s see how that works. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

4. The inverse of a square matrix A is another matrix with the following properties: Here I represents the identity matrix of the same size as A and A-1. Note that A-1 must be a square matrix of the same size as A. Here is a system of linear equations. To solve it, we can put it into matrix format and try to find the inverse of the coefficient matrix. Let’s see how that works. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

5. To find the inverse, for an augmented matrix with the coefficient matrix on the left, and the corresponding identity matrix on the right. Next, row reduce until you have the identity on the left, and the inverse will be on the right. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

6. To find the inverse, for an augmented matrix with the coefficient matrix on the left, and the corresponding identity matrix on the right. Next, row reduce until you have the identity on the left, and the inverse will be on the right. Here is the method, applied to our example: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

7. To find the inverse, for an augmented matrix with the coefficient matrix on the left, and the corresponding identity matrix on the right. Next, row reduce until you have the identity on the left, and the inverse will be on the right. Here is the method, applied to our example: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

8. To find the inverse, for an augmented matrix with the coefficient matrix on the left, and the corresponding identity matrix on the right. Next, row reduce until you have the identity on the left, and the inverse will be on the right. Here is the method, applied to our example: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

9. To find the inverse, for an augmented matrix with the coefficient matrix on the left, and the corresponding identity matrix on the right. Next, row reduce until you have the identity on the left, and the inverse will be on the right. Here is the method, applied to our example: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

10. To find the inverse, for an augmented matrix with the coefficient matrix on the left, and the corresponding identity matrix on the right. Next, row reduce until you have the identity on the left, and the inverse will be on the right. Here is the method, applied to our example: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

11. To find the inverse, for an augmented matrix with the coefficient matrix on the left, and the corresponding identity matrix on the right. Next, row reduce until you have the identity on the left, and the inverse will be on the right. Here is the method, applied to our example: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

12. To find the inverse, for an augmented matrix with the coefficient matrix on the left, and the corresponding identity matrix on the right. Next, row reduce until you have the identity on the left, and the inverse will be on the right. Here is the method, applied to our example: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

13. To find the inverse, for an augmented matrix with the coefficient matrix on the left, and the corresponding identity matrix on the right. Next, row reduce until you have the identity on the left, and the inverse will be on the right. Here is the method, applied to our example: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

14. So now we have the inverse of our coefficient matrix. To solve the original system of equations, simply multiply through by this inverse matrix: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

15. So now we have the inverse of our coefficient matrix. To solve the original system of equations, simply multiply through by this inverse matrix: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

16. So now we have the inverse of our coefficient matrix. To solve the original system of equations, simply multiply through by this inverse matrix: Thus we find a unique solution to the original system of equations. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB