To each square matrix, we associate a number, called its determinant, defined as follows:
If , then det ,
If , then det .
For a square matrix A of dimensions , the determinant can be obtained as follows. First, define as the matrix of dimensions obtained from A by deleting row 1 and column j. Then
det det det det det
Note that, in this formula, the signs alternate between + and -.
For example, if , then
Determinants have several interesting properties. For example, the following statements are equivalent for a square matrix A: