6.2 Properties of Determinants

Finding a determinant using row reductions

One can find the determinant of a matrix by performing row reductions. With the following properties:

1) If B is obtained from A by dividing a row of A by a scalar k then det(B) =1/k(det(A))

OR kdet(B) =det(A)

2) If B is obtained from A by a row swap then det(B) = -det(A)

3) If B is obtained from A by adding a multiple of one row to another row then det(B) = det(A)

Example 1

Find the determinant of the matrix by row reductions

Example 1 Solution

Problems 12 and 14

12 and 14 Solution

Problem 11

Problem 11 Solution

• Det(A) = 8

Det(B) = (8)(-9) = -72

To determine if a matrix is singular

A square matrix is invertible if and only if

Det(A)≠0

A square matrix is singular if and only if

Det(A)=0

Example 3

• Determinant of a productdet(AB) = det(A)*det(B)

Use this fact to show that there is no matrix such that

Example 3 Solution

Determinant of the Transpose of a Matrix

Why is this true?

Homework p.273 1-15 all ,29

Determinant of the inverse of a Matrix