3.6 systems and matrices

3.6 SOLVING SYSTEMS USING MATRICES

MATRICES

A matrix is a rectangular array of numbers, displayed within brackets.

The dimensions of a matrix are the numbers of rows by the numbers of columns in the array.

2 4 1

6 5 3A

MATRICES

Each number in a matrix is a matrix element and can be identified by its row and column number Example:

1311 12

21 22 23

aa aA

a a a

EXAMPLE: IDENTIFYING A MATRIX ELEMENT

What is element in matrix A? 23a

4 9 17 1

0 5 8 6

3 2 10 0

A

SYSTEMS OF EQUATIONS AND MATRICES We can represent systems of equations as

matrices Each row represents an equation Each column represents the coefficients of a variable

Example:3 7 1 3 7

3 8 3 1 8

x y

x y

REPRESENTING SYSTEMS WITH MATRICES

1. Stack the variables/constants2. Write the matrix using coefficients and

constants.• Use a 1 if there is no coeffiecient• Use a 0 if the variable is not included in the

equation

EXAMPLE: REPRESENT THE SYSTEM WITH A MATRIX

3 6

3 12

5 1

x y z

x z

y x

EXAMPLE: WRITE THE SYSTEM OF EQUATIONS REPRESENTED BY THE MATRIX

5 2 7

0 1 9

SOLVING A SYSTEM USING A MATRIX We can solve a system by using a matrix and

performing row operations

Row Operations are the “legal moves and manipulations” we can make in a matrix

Solving a system using row operations is similar to elimination, because we use the same steps, but don’t have variables

SOLVING A SYSTEM USING MATRICES Row Operations:

Switch any two rows Multiply a row by a constant Add one row to another row

SOLVING A SYSTEM USING MATRICES Goal: To use row operations to get a

matrix in the following forms:

Matrices that represent the solution of a system are in reduced row echelon form.

1 0 01 0

0 1 00 1

0 0 1

aa

or bb

c

SOLVE THE SYSTEM OF EQUATIONS USING A MATRIX

4 1

2 5 4

x y

x y

SOLVE THE SYSTEM OF EQUATIONS USING A MATRIX9 2 5

3 7 17

x y

x y

SOLVE THE SYSTEM OF EQUATIONS USING A MATRIX

2 16

3 8

x y

x y