Properties of
Determinants

Recall the definition of a third order
determinant from 5.4:
a1
b1
c1
D  a2
b2
c2
a3
b3
c3

 a1b2 c3  a2b3c1  a3b1c2  a1b3c2  a2b1c3  a3b2 c1
If we rearrange the formula and apply the
distributive property we get the following:
a1b2 c3  a1b3c2  a2b1c3  a2b3c1  a3b1c2  a3b2 c1
 a1 b2 c3  b3c2   a2 b1c3  b3c1  a3 b1c2  b2 c1 

If you look carefully at the parentheses, this can
actually be rewritten again as:
 a1


b2
c2
b3
c3
 a2
b1
c1
b3
c3
 a3
b1
c1
b2
c2
This method is an alternative (but equivalent)
way to find the determinant of a matrix. We call
this method
by
expansion
minors
.
The
minor
of an element in a
determinant is the determinant resulting from the
deletion of the row and column containing the
element.

For example, given the determinant:
4 3 9
2 5 2
7 8 0

Find the minor of 4

Find the minor of 2
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.
2 1 7
5 6 0
4 3 1
row 3
Evaluate the determinant using
expansion by minors.
2.
6 4 0
3 5 3
1 2 0
column 2
*Note: If you aren’t given a row or column
to expand by, choose the row or column
zeros
with the most
to make
it easier!
 We can also expand by minors for larger
order determinants as well using the same
process.

Evaluate the determinant using expansion by minors.
Choose your own row or column!
3.
1 2 3 0
1 1
0 2
0 2
0 3
2
3 4 1
Evaluate the determinant using expansion by minors.
Choose your own row or column!
4.
0 4 0 3
1 1 5 2
1 2 0 6
3 0 0 1