Transcript Chapter 2 Section 1

Chapter 2 Section 1
Solving a System of Linear Equations
(using Matrices)
The Gaussian Elimination Method
• Steps for the Gauss -Jordan elimination
method listed on pages 64 and 63.
• Example 5 on page 57 is an example of
using the Gauss-Jordan elimination method
on a system that has been converted into a
matirx
Convert the system of equations into a
matrix form
• It is important that we first arrange the variables,
and their coefficients, in the same exact order in
each equation on the left side of the equal sign and
have the constants on the right hand side of the
equal sign.
• This is done so that each column will correspond
to a variable (or the constant after the equal sign).
• Example: See Examples: 5 (page 57), 2 (page 65),
3 (page 65), and 5 (page 67).
Objective of the Gaussian Elimination
Method
• To get the matrix in the following form:
1 0 0
0 1 0
0 0 1
*
*
*
using the Three Elementary Row Operations
The Pivot Number/Row
• The circled number is a pivot number and the row
that the pivot number is in is called the pivot row.
• Objective is to change the pivot number to 1 and
to change all the entries above and/or below the
pivot number into a 0 by using the any of the three
elementary row operations.
The First Elementary Row Operations
1. Rearrange the equations in any order.
Notation:
R2
R3
Notation indicate to interchange Row 2
and Row 3
The Second Elementary Row Operation
2. Multiply an equation by a nonzero
number.
Notation: ( – ¼) R2
Notation indicates to multiply – ¼ to all
the coefficient in Row 2.
(This operation is used to change the pivot number to a 1)
The Third Elementary Row Operation
3. Change the equation by adding to it a
multiple of another equation.
Notation:
R2 + ( – 5) R3
Explanation of the notation on the next
slide!
(This operation is used to change values above and below the pivot
number to zero)
Explanation of the Notation for the Third
Elementary Row Operation
•
Notation: R2 + ( – 5) R3
On a scratch piece of paper:
1. Write down the numbers in Row 2
2. Write down the result of the ( – 5) times the numbers
in Row 3 below the corresponding numbers in Row 2.
3. Add the corresponding numbers together.
On the original paper
4. Replace Row 2 (since it is the first row listed in the
notation) by the result of the sum in the previous step
in the matrix
Converting a Pivot Number to a 1
• When converting a pivot number (i.e. the
circled number) to a 1, use the second
elementary row operation.
• Multiply the pivot row by the reciprocal of
the circled number.
Converting Any Number Above or
Below the Pivot Number to a 0
• When converting a number, above or below the
pivot number, to a zero, use the third elementary
row operation.
• The row in which the number that you want to
change to a zero, is listed first in the notation of
the third elementary row operation.
• Take the number that you want to change to a
zero, change the sign, and the resulting number is
the number that you want to multiply to the pivot
row (the second part of the third elementary row
operation).
Examples
• Example 5 on page 57
• Example 4 on page 56 (but converted into
matrix form)
• Other examples.
Matrices on the Calculator
• See Appendix B page A10 – A11 for entering, editing,
and erasing matrices.
• We will use the rref function in the calculator to
perform the Gaussian Elimination Method on a
matrix.
Get into the matrix mode ( [2nd] [x – 1/MATRIX] )
Select MATH
Select B:rref( and hit ENTER
Get back into the matrix mode and select the matrix you
want to perform the Gaussian elimination method on
– Hit enter and the matrix that appears on the calculator is the
matrix that is the result of performing the Gaussian
elimination method.
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