Properties of Determinants

Recall the definition of a third order determinant from 5.4:

If we rearrange the formula and apply the distributive property we get the following:

D a1 b1 c1

a2 b2 c2

a3 b3 c3

a1b2c3 a2b3c1 a3b1c2 a1b3c2 a2b1c3 a3b2c1

a1b2c3 a1b3c2 a2b1c3 a2b3c1 a3b1c2 a3b2c1

a1 b2c3 b3c2 a2 b1c3 b3c1 a3 b1c2 b2c1

If you look carefully at the parentheses, this can actually be rewritten again as:

This method is an alternative (but equivalent) way to find the determinant of a matrix. We call this method by

. The of an element in a determinant

is the determinant resulting from the deletion of the row and column containing the element.

a1

b2 c2

b3 c3

a2

b1 c1

b3 c3

a3

b1 c1

b2 c2

expansion

minors

minor

For example, given the determinant:

Find the minor of 4

Find the minor of 2

4 3 92 5 27 8 0

Expansion by Minors 1. Determine the column or row to be expanded by.

(either given or chosen) 2. The signs in front of the terms follow this pattern:

to determine the signs on the terms, you can add the row # and column # of the first term: if it is even, start with a +, if it is odd, start with a – and alternate signs.

3. Lay out the terms and blank second order determinants with the correct signs.

4. Fill in the second order determinants by finding the minor of the term in front of the determinant.

5. Evaluate the second order determinants and simplify to find the determinant of the third order determinant.

Evaluate the determinant using expansion by minors.

1.

row 3

2 1 7 5 6 0 4 3 1

Evaluate the determinant using expansion by minors.

2.

column 2

6 4 0 3 5 31 2 0

*Note: If you aren’t given a row or column to expand by, choose the row or column with the most to make it easier!

We can also expand by minors for larger order determinants as well using the same process.

zeros

Evaluate the determinant using expansion by minors. Choose your own row or column!

3. 1 2 3 0 1 1 0 20 2 0 32 3 4 1

Evaluate the determinant using expansion by minors. Choose your own row or column!

4. 0 4 0 3 1 1 5 21 2 0 63 0 0 1