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13
1 Chapter Content Determinants by Cofactor Expansion Evaluating Determinants by Row Reduction Properties of the Determinant Function A Combinatorial Approach to Determinants
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Chapter Content. Determinants by Cofactor Expansion Evaluating Determinants by Row Reduction Properties of the Determinant Function A Combinatorial Approach to Determinants. Theorems. Theorem 2.2.1 Let A be a square matrix - PowerPoint PPT Presentation
Transcript of Chapter Content
1
Chapter Content
Determinants by Cofactor Expansion Evaluating Determinants by Row Reduction Properties of the Determinant Function A Combinatorial Approach to Determinants
2
Theorems Theorem 2.2.1
Let A be a square matrix If A has a row of zeros or a column of zeros, then det(A)
= 0.
Theorem 2.2.2 Let A be a square matrix det(A) = det(AT)
3
Theorem 2.2.3 (Elementary Row Operations) Let A be an nn matrix
If B is the matrix that results when a single row or single column of A is multiplied by a scalar k, than det(B) = k det(A)
If B is the matrix that results when two rows or two columns of A are interchanged, then det(B) = - det(A)
If B is the matrix that results when a multiple of one row of A is added to another row or when a multiple column is added to another column, then det(B) = det(A)
Example 1
333231
232221
131211
k k k
)det(
aaa
aaa
aaa
B
4
2-2 Example of Theorem 2.2.3
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Theorem 2.2.4 (Elementary Matrices) Let E be an nn elementary matrix
If E results from multiplying a row of In by k, then det(E) = k
If E results from interchanging two rows of In, then det(E) = -1
If E results from adding a multiple of one row of In to another, then det(E) = 1
Example 2
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Theorem 2.2.5 (Matrices with Proportional Rows or Columns) If A is a square matrix with two proportional rows or two
proportional column, then det(A) = 0
Example 3
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2-2 Example 4 (Using Row Reduction to Evaluate a Determinant) Evaluate det(A) where
Solution:
162
9 63
510
-
A
.
1 6 2
5 1 0
3 2 1
3
1 6 2
5 1 0
9 6 3
1 6 2
9 6 3
5 1 0
)det(
A
The first and second rows of A are interchanged.
A common factor of 3 from the first row was taken through the determinant sign
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2-2 Example 4 (continue)
.
165)1)(55)(3(
1 0 0
5 1 0
3 2- 1
)55)(3(
55- 0 0
5 1 0
3 2- 1
3
5- 10 0
5 1 0
3 2- 1
3
-2 times the first row was added to the third row.
-10 times the second row was added to the third row
A common factor of -55 from the last row was taken through the determinant sign.
1 6 2
5 1 0
3 2 1
3)det(
A
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2-2 Example 5 Using column operation to evaluate a determinant
Compute the determinant of
5-137
0360
6072
3001
A
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2-2 Example 6 Row operations and cofactor expansion
Compute the determinant of
3573
5142
11-21
62-53
A
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Exercise Set 2.2 Question 12
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Exercise Set 2.2 Question 13
13
Exercise Set 2.2 Question 20
