Algebra 2

Monday 11-17-14
Warm ups
◦ Complete Slide below

Discussion/Notes/Guided Practice
◦ 4.5 Determinants

Assignment:
◦ A#4.5 page 198 #10-30 evens
Today, you will be able to:

Define new vocabulary: determinant, second
order determinant, third-order determinant,
expansion by minors

Evaluate the determinant of a 2x2 matrix

Evaluate the determinant of a 3x3 matrix
I-Ready Testing
Thursday
Success Criteria:
Q&A
 Guided Practice
 Homework

Prior Knowledge Self-Evaluation – Rate each learning target:
1.
Define new vocabulary:
determinant, second order
determinant, third-order
determinant, expansion by
minors
2.
Evaluate the determinant
of a 2x2 matrix
3.
Evaluate the determinant
of a 3x3 matrix
Determinants

A determinant is a special number that can be
calculated from a square matrix.
What is it for???
The determinant tells us things about the matrix that are useful
in:
◦ systems of linear equations,
◦ calculus
◦ and helps us find the inverse of a matrix
Second-order determinant: __________________________________
Third-order determinant: ____________________________________
Determinants – Second Order
Words:
The value of a second-order determinant is found by
calculating the difference of the products of the two
diagonals.
Symbols:
𝑎
𝑐
𝑏
= 𝑎𝑑 − 𝑏𝑐
𝑑
Example:
3 −1
= 3 5 − 2 −1 = 17
2 5
Example #1: 2nd order Determinants
1. Find the value of the determinant
6 4
−1 0
2. Find the value of the determinant
7 4
−3 2
3. Find the value of the determinant
−4 6
−3 −2
Determinants – Third Order
Third-Order determinants can be calculated by two methods:
1. Expansion by Minors
2. Diagonals
The minor of an element is the determinant formed when the row and
column containing that element are deleted.
𝑎
𝑑
𝑔
𝑏
𝑒
ℎ
𝑐
𝑓
𝑖
the minor of a is
𝑎
𝑑
𝑔
𝑏
𝑒
ℎ
𝑐
𝑓
𝑖
the minor of b is
𝑎
𝑑
𝑔
𝑏
𝑒
ℎ
𝑐
𝑓
𝑖
the minor of c is
Determinants – Third Order
Expansion by Minors
To use expansion by minors with 3rd order determinants, each member of one row is
multiplied by its minor and its position sign, and the results are added together.
Position Sign
𝑎
𝑑
𝑔
𝑏
𝑒
ℎ
𝑐
𝑓 =𝑎 𝑒
ℎ
𝑖
+
−
+
− +
+ −
− +
𝑑
𝑓
−𝑏
𝑔
𝑖
𝑓
𝑑
+𝑐
𝑔
𝑖
Example: Evaluate using expansion by minors.
1 0
2 −1
4 −2
−1
3
−3
𝑒
ℎ
Determinants – Third Order
Position Sign
+
−
+
− +
+ −
− +
𝑎
𝑑
𝑔
𝑏
𝑒
ℎ
𝑐
𝑓 =𝑎 𝑒
ℎ
𝑖
𝑑
𝑓
−𝑏
𝑔
𝑖
Practice: Evaluate using expansion by minors.
−2
5
4
3
−3
−6
−1
8
−5
𝑓
𝑑
+𝑐
𝑔
𝑖
𝑒
ℎ
Determinants – Third Order
Using Diagnonals
Step 1: Write the first two columns on the right side of the determinant.
𝑎
𝑑
𝑔
𝑏
𝑒
ℎ
𝑐
𝑓
𝑖
Step 2A: Draw diagonals from each element of the top row of the determinant
downward to the right. Find the product of the elements on each diagonal.
𝑎
𝑑
𝑔
𝑏
𝑒
ℎ
𝑐 𝑎
𝑓 𝑑
𝑖 𝑔
𝑏
𝑒
ℎ
Determinants – Third Order
Using Diagonals
Step 2B: Draw diagonals from each element of the third row of the determinant
upward to the right. Find the product of the elements on each diagonal.
𝑎
𝑑
𝑔
𝑏
𝑒
ℎ
𝑐 𝑎
𝑓 𝑑
𝑖 𝑔
𝑏
𝑒
ℎ
Step 3: Add the products of the first set of diagonals and then subtract the products
of the second set of diagonals.
𝑎𝑒𝑖 + 𝑏𝑓𝑔 + 𝑐𝑑ℎ − 𝑔𝑒𝑐 − ℎ𝑓𝑎 − 𝑖𝑑𝑏
Determinants – Third Order
EXAMPLE: Using Diagonals
Evaluate using diagonals:
3
2
1
−2 −1
−1 0
2 −3
Determinants – Third Order
PRACTICE: Using Diagonals
Evaluate using diagonals:
1
0
5
−5 3
2 −7
−1 −2
Self-Evaluation - Rate each learning target:
Define new vocabulary: determinant, second order determinant,
third-order determinant, expansion by minors
4
3
2
1
0
1
0
1
0
Evaluate the determinant of a 2x2 matrix
4
3
2
Evaluate the determinant of a 3x3 matrix
4
3
2
Assignment:
A#4.5 page 198 #10-30 evens