For each square matrix there is a number that is called its determinant.
The determinant is characterized by the following properties:
The determinant of a ``triangular'' matrix is the product of its diagonal entries.
The determinant of a matrix changes sign if two rows are switched.
If a row is multiplied by a non-zero constant, then the determinant is multiplied by that same constant.
If a row is replaced by its sum with a muliple of another row, the determinant remains unchanged.
A square matrix is invertible if and only if its determinant is non-zero.
The determinant of the product of two matrices is the product of their determinants.
Monday, March 15, will be a day of review. Please bring specific questions.
Find the determinant of the 4 \times 4 matrix
| 1 | 1 | -2 | -1 |
| 3 | -5 | -4 | 3 |
| 2 | -2 | 3 | 1 |
| 1 | 2 | -1 | 0 |
Try to find a redundancy in the list of 4 rules above that were said to characterize the determinant.
Use the rules above to conclude that the determinant of the 2 \times 2 matrix
| a | b |
| c | d |
is the number a d - b c .
Use the rules above to find the determinant of the 3 \times 3 matrix
| a | b | c |
| d | e | f |
| 0 | 0 | g |