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Determinants: Row Operations and Properties

Explore the key theorems, properties, and operations of determinants on matrices at Appalachian State University. Learn how row and column manipulations affect determinants and uncover new insights into matrix simplification and invertibility. Dive into determinant properties and essential homework exercises.

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Determinants: Row Operations and Properties

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  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

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