Transcript Lesson 13-1: Matrices & Systems

Lesson 13-1: Matrices &
Systems
Objective:
Students will:
State the dimensions of a matrix
Solve systems using matrices
Matrix
An array of numbers aligned in rows & columns (rows are always
first)
 a11 a12  a1n 
a

a

a
21
22
2n 

 


 


am1 am 2  amn 
a11 = number in 1st row, 1st column
Dimensions
Stated as “rows Χ columns” (rows are always first)
5  1
 2 0 6
 3 1  8 3 12 


  1 5 7  2 8 
Dimensions:
3X5
Using Matrices to Solve Systems
► Use Standard Form: Ax + By = C
► Put coefficients and constant in a matrix
► Solve by getting zeros in the lower left diagonal
- 0’s below the main diagonal is called triangular form
► Allowable operations
- interchanging 2 rows
- multiplying a row by a constant (not zero)
- adding 2 rows replacing a row with the result
► Write equations in x-y form from triangular form
► Solve – substitute - solve
Example 1
Solve
x + 2y = 3
3x + 8y = 1
x’s
y’s
1 2 3


3
8
1


●-3
  3  6  9


3
8
1


add &
replace
2nd row
  3  6  9


0
2

8


Main
diagonal
  3  6  9


0
2

8


Triangular
form
-3x -6(-4) = -9
2y = -8
-3x +24 =-9
y = -4
x = 11
Now use this
Now we are
done!!!
Since we are down
to 1 variable and
answer we can
convert back and
solve
Example 2
Solve 2x + 4y + 8z = 6
x + 3y + 5z = 4
3x + 8y + 6z = 19
4
8
6  ●3
2
 3  9  15  12

 ●-2
 0  1  9
7 
x
y
z
2 4 8 6 
1 3 5 4 


3 8 6 19
●-3
Then add 2 and
3 replace 3
Then add 1and
2 replace 2
6 12 24 18 
Then add 2 and
0  6  6  6
3 replace 3


0  1  9 7  ●-6
Since we are down
to 1 variable and
answer we can
convert back and
solve
6 12 24 18 
0  6  6  6 


0 0 48  48
Cont…..
6 12 24 18 
0  6  6  6 


0 0 48  48
6x +12(2)+24(-1)= 18
48z = -48
z = -1
-6y -6(-1) = -6
6x =18
-6y = -12
x=3
y=2
Now we are done!!!
Now use this
Now use
this
Remember- You can put these back into the
original 3 equations to make sure they are the
solution to all of them. I know I did!!!
You try:
• 5x - 2y = -44
• x + 5y = 2
3X3
x - 2y + 3z = 4
2x - y + z = -1
4x + y + z = 1
Assignment
• 13-1/572/1-13 odd