Determinant of matrix

Definition

Determinant is a value can be calculate from a square matrix.

For A is a matrix,

For A is a matrix,

So for A is a matrix,
Do that for the i-th row:

or do that for the j-th column:

Where is Cofactor and is Minor.

Properties of determinants

1. |A| = |A'|
2. Interchange of any two rows will alter the sign
3. The determinant of multiplication of any row/ column of A by scalar k will be k|A|
4. The determinant of multiplication of the whole matrix A by scalar k will be
5. If one row / column is a multiple of another row/column, |A| = 0.
6. If one row / column are all 0, |A| = 0
7. 

(It is very useful to separate matrices to those with only one non-zero value in each row / column)

8. |AB| = |A||B|
9. || = 1/|A|
10. If c is a scalar, then |cA| =
11. If A is lower or upper triangular, |A| = . In particular, || = 1.

Note

Example