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

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

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.

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.

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.

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 = 0 1w + 4y = 0 1x + 4z = 1

12-4: Matrix Methods for Square Systems Solve the systems 2w + 6y = 1 2x + 6z = 0 1w + 4y = 0 1x + 4z = 1 2w + 6y = 1 2x + 6z = 0 -2w – 8y = 0 -2x – 8z = -2 -2y = 1 -2z = -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

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

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

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

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

12-4: Matrix Methods for Square Systems Assignment Page 819 14 – 34, evens