10.3 Day 2 Systems of Linear Equations ­ Determinants 2011
January 10, 2011
10.3 Systems of Linear Equations: Determinants DAY 2
Objective:
Evaluate 3 by 3 determinants.
Jan 8­12:33 PM
1
10.3 Day 2 Systems of Linear Equations ­ Determinants 2011
January 10, 2011
Warm­up:
Solve the following system using Cramer's Rule.
1. 5x ­ y = 13
{2x + 3y = 12
{
ax + by = s
cx + dy = t
Dx
x = D = s b
t d
a b
c d
Dy
y = D = a s
c t
a b
c d
Jan 8­4:31 PM
2
10.3 Day 2 Systems of Linear Equations ­ Determinants 2011
January 10, 2011
Determinants
This is a 3 by 3 determinant.
a11 a12 a13
D = a21 a22 a23
a31 a32 a33
OK we know what 3 by 3 determinant means but what do those little numbers mean?
Jan 8­4:37 PM
3
10.3 Day 2 Systems of Linear Equations ­ Determinants 2011
January 10, 2011
Determinants
To find the value of a 3 by 3 determinant you expand across a row or column.
You may choose any row or column. Here are some examples.
Example #1: Expanding down column 2 to evaluate the determinant.
a11 a12 a13
a21 a22 a23
a31 a32 a33
= (­1)1 + 2 a12 a21 a23 + (­1)2 + 2 a22 a a + (­1)3 + 2 a32 a a 11
a31 a33 13
a31 a33 11
13
a21 a23 Jan 8­4:45 PM
4
10.3 Day 2 Systems of Linear Equations ­ Determinants 2011
January 10, 2011
Determinants
To find the value of a 3 by 3 determinant you expand across a row or column.
You may choose any row or column. Here are some examples.
Example #2: Expanding across row 3 to evaluate the determinant.
a11 a12 a13
a21 a22 a23
a31 a32 a33
= (­1)3 + 1 a31 a12 a13 + (­1)3 + 2 a32 a11 a13 + (­1)3 + 3 a33 a a 11
a22 a23 a21 a23 12
a21 a22 Jan 8­4:45 PM
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10.3 Day 2 Systems of Linear Equations ­ Determinants 2011
January 10, 2011
Find the value of the determinant.
Example #3:
3 4 ­1
4 6 2
8 ­2 3
Answer: 138
Jan 8­6:01 PM
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10.3 Day 2 Systems of Linear Equations ­ Determinants 2011
January 10, 2011
Find the value of the determinant.
Example #4:
­2 3 1
5 3 0
4 ­1 6
Jan 8­6:01 PM
7
10.3 Day 2 Systems of Linear Equations ­ Determinants 2011
January 10, 2011
Homework:
page 767 (11 ­ 14)
Jan 8­5:55 PM
8
10.3 Day 2 Systems of Linear Equations ­ Determinants 2011
January 10, 2011
Jan 10­12:53 PM
9