Chapter Content n n Determinants by Cofactor Expansion

Chapter Content n n Determinants by Cofactor Expansion Evaluating Determinants by Row Reduction Properties of the Determinant Function A Combinatorial Approach to Determinants 2021/9/8 Elementary Linear Algebra 1

Theorems n Theorem 2. 2. 1 q Let A be a square matrix n If A has a row of zeros or a column of zeros, then det(A) = 0. n 2021/9/8 Theorem 2. 2. 2 n Let A be a square matrix q det(A) = det(AT) Elementary Linear Algebra 2

Theorem 2. 2. 3 (Elementary Row Operations) n Let A be an n n matrix q q q n 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 2021/9/8 Elementary Linear Algebra 3

2 -2 Example of Theorem 2. 2. 3 2021/9/8 Elementary Linear Algebra 4

Theorem 2. 2. 4 (Elementary Matrices) n Let E be an n n elementary matrix q q q n 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 2021/9/8 Elementary Linear Algebra 5

Theorem 2. 2. 5 (Matrices with Proportional Rows or Columns) n If A is a square matrix with two proportional rows or two proportional column, then det(A) = 0 n Example 3 2021/9/8 Elementary Linear Algebra 6

2 -2 Example 4 (Using Row Reduction to Evaluate a Determinant) n Evaluate det(A) where n Solution: The first and second rows of A are interchanged. A common factor of 3 from the first row was taken through the determinant sign 2021/9/8 Elementary Linear Algebra 7

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 row A common factor of -55 from the last row was taken through the determinant sign. 2021/9/8 Elementary Linear Algebra 8

2 -2 Example 5 n Using column operation to evaluate a determinant q 2021/9/8 Compute the determinant of Elementary Linear Algebra 9

2 -2 Example 6 n Row operations and cofactor expansion q 2021/9/8 Compute the determinant of Elementary Linear Algebra 10