| a b | determinant
= ad - bc
| c d |
Example:
|2 6|
|3 4| determinant = 2(4) - 6(3)
= 8 - 18 = -10
II. The determinant of a 3 by 3 matrix.
| 1 0 2|
|-3 4 -1|
| 6 1 -2|
A. Diagonal Method: On
the right side, add the first two columns of the matrix. Multiply
each diagonal
from right to left and add the answers; multiply each diagonal from left
to
right and add
the answers. Subtract the second sum from the second.
|
1 0 2| 1 0
|-3 4 -1| -3 4
| 6 1 -2| 6 1
= [1(4)(-2) + (0)(-1)(6) + 2(-3)(1)] - [6(4)(2) + 1(-1)(1) + (-2)(-3)(0)]
= [ -8 + 0 +(-6)] - [ 48 + (-1) + 0]
= -14 - 47
= -61
B. Method of expansion by minors:
1. Choose
a row or column to use as multipliers. These are called the cofactors.
The
signs of the cofactors are determined by adding the row number to the column
number
and determining if that answer is even or odd. If it is even, put
a + in front of the
cofactor. If it is odd, put a - in front of the cofactor.
2. Cross
out the row and column the first number is in. Write down the elements
that are
still showing. Find the determinant of this smaller matrix and multiply
it by the crossed
out cofactor. Do this for each element in the row or column you chose.
| 1 -2 0 3|
|-3 4 1 5|
Choose row 4. (This is arbitrary) The
idea is to choose the
| 1 2 3 4|
row or column with the simplest multipliers.
|-1 -2 3 0|
-1 | -2 0 3 | + (-2) | 1 0 3 |
- (3) | 1 -2 3 | + (0) | 1 -2
0 |
| 4 1 5 |
| -3 1 5 |
| -3 4 5 |
| -3 4 1 |
| 2 3 4 |
| 1 3 4 |
| 1 2 4 |
| 1 2 3 |
-1 [(-8 + 0 + 36) - (6 - 30 + 0)] - 2 [(4 + 0 - 27) - (3 + 15 - 0)]
- 3 [(16 -10 -18) - (12 + 10 + 24)] + 0
= -1 [28 - (-24)] - 2(-23 - 18) - 3 (-12 - 46) + 0
= -1 (52) -2 (-41) - 3(-58) + 0
= -52 + 82 + 174 + 0
= 204
| 3 0 0 0 0|
| 4 1 0 0 0|
|-1 2 3 0 0|
| 5 1 -2 2 0|
| 3 4 1 1 1|
The determinant is the product of the entries on the main diagonal (left top to right bottom).
This determinant is 18.
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