3.6 systems and matrices
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Transcript of 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