1. 12-4: Matrix Methods for Square Systems
Is this the end???

2. 12-4: Matrix Methods for Square Systems
Square System: A matrix that has the same number of equations as variables.
Instead of using a single matrix to represent the system, we can break the system into three parts: the coefficients, the variables, and the constants.
coefficients variables constants

3. 12-4: Matrix Methods for Square Systems
Note that for the system above: AX = B
To solve a matrix equation AX = B, it is necessary to “undo” the multiplication. But that relies on the fact that there is a matrix such that [A][A-1] = 1
Thus, in order to define the inverse of a matrix, we must first define the identity for matrix multiplication.

4. 12-4: Matrix Methods for Square Systems
The n x n identity matrix In is the matrix with n rows and n columns that has 1’s on the diagonal from the top left to bottom right, and 0’s as all other entries.
The identity matrix is like multiplying a number by 1. It returns the original matrix. Order doesn’t matter.

5. 12-4: Matrix Methods for Square Systems
An n x n matrix is considered invertible if there exists an n x n matrix B such that AB = In
The matrix B is called the inverse of A, and is written as A-1, where AA-1 = In. Not all matrices have multiplicative inverses.
Example 3: Verifying an inverse.
For the given matrices, verify B is the inverse of A.

6. 12-4: Matrix Methods for Square Systems
There are several methods for finding the inverse of an invertible matrix.
Method 1: Using variables
Find the inverse of
Suppose Then AA-1 = I2
So you have 2w + 6y = 1 2x + 6z = w + 4y = 0 1x + 4z = 1

7. 12-4: Matrix Methods for Square Systems
Solve the systems 2w + 6y = 1 2x + 6z = w + 4y = 0 1x + 4z = 1
2w + 6y = 1 2x + 6z = 0 -2w – 8y = 0 -2x – 8z = -2
-2y = z = -2
y = -½ z = 1
2w + 6(-½) = 1 2x + 6(1) = 0
2w – 3 = 1 2x + 6 = 0
2w = 4 2x = -6
w = 2 x = -3
A-1 therefore is

8. 12-4: Matrix Methods for Square Systems
You can use your calculator to create the inverse matrix.
Create the original matrix
Enter A-1, where A is your original matrix
Example in class using the previous problem

9. 12-4: Matrix Methods for Square Systems
Solving square systems using inverse matrices
If AX = B
A is matrix of coefficients
X is matrix of variables
B is matrix of constants
Then:
AA-1X = A-1B
InX = A-1B
X = A-1B, which gives us a solution to our variables

10. 12-4: Matrix Methods for Square Systems
Solving square systems using inverse matrices
Use an inverse matrix to solve
Turn the system into two matrices (coefficient and solution matrices)
(using calculator)
So x = 3 & y = -1

11. 12-4: Matrix Methods for Square Systems
Use an inverse matrix to solve

12. 12-4: Matrix Methods for Square Systems
Assignment
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