MATRICES AND LINEAR SYSTEMS

Mr. Velazquez

Honors Precalculus

LINEAR SYSTEMS WITH MATRICES

The 12 numbers inside the brackets are arranged in two rows and six columns. This rectangular array of 12 numbers, arranged in rows and columns and placed in brackets, is an example of a matrix (plural: matrices).The numbers inside the brackets are called elements of the matrix. Matrices are used to display information and to solve systems of linear equations. Because systems involving two equations in two variables can easily be solved by substitution or addition, we will focus on matrix solutions to systems of linear equations in three or more variables.

LINEAR SYSTEMS WITH MATRICES

Main Diagonal

Notice how the second matrix contains 1s down the diagonal from the upper left to the lower right (called the main diagonal) and 0s below the 1s. This arrangement is called row-echelon form and makes it easy to find the solution of the system of equations with just a little back-substitution.

LINEAR SYSTEMS WITH MATRICESWrite the augmented matrix of the given system of equations.

3x+4y=7

4x-2y=5

MATRIX ROW OPERATIONS

GAUSSIAN ELIMINATION

GAUSSIAN ELIMINATION

2 1 2

Write the system of equations corresponding to this

augmented matrix. Then perform the row operation

on the given augmented matrix. R 4

1 3 3 5

4 5 3 5

3 2 4 6

r r= +

− − − − − − − −

GAUSSIAN ELIMINATIONUse Gaussian Elimination to solve the following system:

ቐ2𝑥 + 4𝑦 − 2𝑧 = 16

𝑥 − 𝑧 = 22𝑦 + 3𝑧 = −3

GAUSS-JORDAN ELIMINATION

With Gauss-Jordan Elimination, we are trying to re-write the matrix into reduced row-echelon form, shown below:

USING MATRICES WITH CALCULATORS

To work with matrices we need to press the keys to get to the Matrix menu (for most TI models).

Move the cursor right twice to get to EDIT, press ENTER and key in the dimensions of matrix A, pressing ENTER after each number. To get out of the matrix menu, press QUIT (2nd, MODE)

To get to reduced row-echelon form, bring up the Matrix menu and move the cursor right once to MATH, then scroll down to B: rref(, and press ENTER.

To enter the name of a matrix you have already defined, bring up the Matrix menu again and use the NAMES

EXAMPLE:

Solve this system of equations using matrices (row operations).

3 5 3

15 5 21

x y

x y

− =

+ =

EXAMPLE:

Solve this system of equations using matrices (row operations).

2 4

2 4 0

3 2 11

x y

y z

x z

+ = −

− + =

− = −

CLASSWORK & HOMEWORK

CLASSWORK: MATRICES & LINEAR SYSTEMS

(Due at the end of class today) Solve the system below using either Gaussian or Gauss-Jordan Elimination. Show all your steps! (No matrix, no credit)

ቐ

2𝑥 + 𝑦 + 2𝑧 = 18𝑥 − 𝑦 + 2𝑧 = 9𝑥 + 2𝑦 − 𝑧 = 6

HOMEWORK:

Systems of Equations HW 3

(Math XL)