Transpose and Determinant

Transpose

Determinant

Determinant

The determinant of a square matrix is where is symmetric group, is a permutation, is the sign of the permutation, and denotes the entry of .

Cofactor Expansion

For any , , where , and denotes the matrix obtained from by removing the th row and th column.

e.g. For a matrix , the determinant can be recursively computed as follows:

Proposition

Determinant satisfies the following properties:

• .
• If is obtained from by scaling a single row by a factor , then . In particular, .
• If all rows of , , except the th are same, and for any , , then .
• If is obtained by swapping 2 rows of , then .
• If is obtained by adding to for , then .
• .