1. Solving Systems Using Matrices
Inverse Matrices

2. Preview Standards and Objectives Defining a Matrix
Writing Systems as Matrices
Solving a System by the Matrix Equation
Why This New Method?
Practice

3. Standards and Objectives

4. Defining a Matrix A matrix is an array or ordered set of numbers
Each matrix has a name, given by a capital letter such as A
A matrix is “classified” by its number of rows and columns…in that order
Each number in a matrix has an “address”

5. Example is a 2x3, read “2 by 3”, matrix named A
Each number is addressed by a lowercase a followed by its row and column
a21 is the number that is in row 2, column 1

6. Matrices and Systems Matrix A: the coefficients from the system
Matrix X: 1 column matrix with first variable on top, going down
Matrix B: 1 column matrix with the constants on the right side of the equal sign

7. Write the matrices for the systems

8. A word of warning Notice the x’s and y’s aren’t on the same side
Each system must be in “standard form” of
Ax + By = C
Rewrite the system before writing the matrices

9. The Matrix Equation: AX=B
A X = B A-1A X = B A-1 X = B A-1
Matrix A times Matrix X equals Matrix B
To solve for matrix X, we use the “inverse matrix” A-1
We will use the calculator to do the calculation part

10. Write out the other examples using the matrix equation

11. Entering the Matrix Enter Matrix A Enter Matrix B MENU MAT for Matrix
Select Mat A and press right arrow on D-Pad
Give dimensions of Matrix
2 ENTER, 2 ENTER
Input coefficients into matrix
5 ENTER, 3 ENTER, 3 ENTER, 2 ENTER
Press Exit
Select Mat B and press right arrow on D-Pad
Give dimensions of Matrix
2 ENTER, 1 ENTER
Input constants into matrix
1 ENTER, -3 ENTER

12. Solving on the Calculator
Go to RUN
Press OPTN button (next to shift)
F2 for MAT (matrix)
F1- MAT again (puts a Mat on the screen)
ALPHA A
SHIFT x-1 (this gives us the inverse of A)
F1- MAT again
ALPHA B
Press EXE

13. Reading the Solution
The matrix it gives you as the answer is the x and y values of the system
If you get a Ma Error (Math Error)
Could be no solution
Could be infinite solutions
You will have to solve by hand to figure out which is which

14. Try Solving the Other Examples

15. Why this new method?
Tomorrow we will do this all again with systems that have 3 variables.
Imagine doing substitution and elimination with 3 or more different equations.
It is possible, but it takes some time…
For now, let’s practice 2 variable systems

16. Practice Write each system as a matrix and use the matrix equation
A X = B
Show the steps:
X = B A-1
Use the calculator to solve the matrix equation
p.146 #’s 28-30, 37-42
p.147 #50 & 52