1 Chapter Content Determinants by Cofactor Expansion Evaluating Determinants by Row Reduction Properties of the Determinant Function A Combinatorial Approach to Determinants 2014119 Elementary Linear Algebra ID: 759473
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Slide1
2014/1/19
Elementary Linear Algebra
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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
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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(
A
T
)
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Theorem 2.2.3 (Elementary Row Operations)
Let A be an nn matrixIf 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
Slide42-2 Example of Theorem 2.2.3
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Theorem 2.2.4 (Elementary Matrices)
Let E be an nn elementary matrixIf E results from multiplying a row of In by k, then det(E) = kIf E results from interchanging two rows of In, then det(E) = -1If E results from adding a multiple of one row of In to another, then det(E) = 1Example 2
Slide6Theorem 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) = 0Example 3
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2-2 Example 4 (Using Row Reduction to Evaluate a Determinant)
Evaluate det(A) whereSolution:
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)
-2 times the first row was added to the third row.
-10 times the second row was added to the third rowA common factor of -55 from the last row was taken through the determinant sign.
Slide92-2 Example 5
Using column operation to evaluate a determinant Compute the determinant of
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Slide102-2 Example 6
Row operations and cofactor expansionCompute the determinant of
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