Determinants: Row Operations and Properties

1. Determinants:Row Operations andProperties Mark Ginn Math 2240 Appalachian State University

2. Theorem: Let A and B be square matrices: 1. If B is obtained from A by interchanging 2 rows then det(A) = -det(B) 2. If B is obtained from A by adding a multiple of one row to another row then, det(A) = det(B) 3. If B is obtained from A by multiplying a row of A by a constant c then, c *det(A) = det(B)

3. Another way to simplify matrices • Fact: When elementary column operations are applied to a matrix A to obtain a matrix B, they have the same effect on the determinant as row operations.

4. Theorem: Let A and B be square matrices: 1. If B is obtained from A by interchanging 2 columns then det(A) = -det(B) 2. If B is obtained from A by adding a multiple of one column to another column then, det(A) = det(B) 3. If B is obtained from A by multiplying a column of A by a constant c then, c *det(A) = det(B)

5. Some new thoughts about determinants • It is now easy to show that any matrix that is row equivalent to the identity matrix has a nonzero determinant. • This also implies that a matrix has determinant 0 if and only if it is row equivalent to a matrix with an all 0 row. • We now get the following extension to THE BIG THEOREM

6. THE NEW BIG THEOREM If A is an n by n matrix, then TFSAE. 1. A is invertible. 2. Ax = b has a unique solution for all b. 3. Ax = 0 has only the trivial solution. 4. A is row equivalent to In. 5. A can be written as the product of elementary matrices. 6. det(A) ≠ 0.

7. Some other properties of determinants 1. det(AB) = det(A) *det(B). 2. det(cA) = cndet(A) 3. If A is invertible then det(A-1) = 1/det(A). 4. det(A) = det(AT).

8. Homework • p. 125: 15,17,19,28,29-34,37,38 • p. 135: 6,10,11,18,2437,38,39