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

2. Here I represents the identity matrix of the same size as A and A-1.
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. Here I represents the identity matrix of the same size as A and A-1.
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. Here I represents the identity matrix of the same size as A and A-1.
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. Here is the method, applied to our example:
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. Here is the method, applied to our example:
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. Here is the method, applied to our example:
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. Here is the method, applied to our example:
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. Here is the method, applied to our example:
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. Here is the method, applied to our example:
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. Here is the method, applied to our example:
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. Here is the method, applied to our example:
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
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
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. Thus we find a unique solution to the original system of equations.
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

17. 2x2 Inverse Matrix Shortcut
Theorem 4: Let If , then
A is invertible and
If , then A is not invertible.
The quantity is called the determinant of A, and we write
This theorem says that a matrix A is invertible if and only if det

18. Here I represents the identity matrix of the same size as A and A-1.
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. Notice that the coefficient matrix is the same as the one we solved earlier. We can use the same inverse matrix to solve this one.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

19. Here I represents the identity matrix of the same size as A and A-1.
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. Notice that the coefficient matrix is the same as the one we solved earlier. We can use the same inverse matrix to solve this one.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

20. A is an invertible matrix. A is row equivalent to the identity matrix.
Theorem 8: Let A be a square matrix. Then the following statements are equivalent. That is, for a given A, the statements are either all true or all false.
A is an invertible matrix.
A is row equivalent to the identity matrix.
A has n pivot positions.
The equation has only the trivial solution.
The columns of A form a linearly independent set.
The linear transformation is one-to-one.
The equation has at least one solution for each b in Rn .
The columns of A span Rn .
The linear transformation maps Rn onto Rn .
There is an matrix C such that
There is an matrix D such that
AT is an invertible matrix.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB