Solving Systems Using Matrices Inverse Matrices

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

Standards and Objectives

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”

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

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

Write the matrices for the systems

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

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

Write out the other examples using the matrix equation

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

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

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

Try Solving the Other Examples

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

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