1. Determinants King Saud University

2. The inverse of a 2 x 2 matrix Recall that earlier we noticed that for a 2x2 matrix,

3. From this we deduced that a 2x2 matrix A is singular if and only if ad-bc = 0. This quantity (ad-bc) has some other useful properties as well and so is defined to be the determinant of the matrix A.

4. Determinants of larger matrices As we noted earlier, there is no “nice” formula for the inverse of larger than 2x2 matrices. We still can define the determinant of a larger square matrix and it will still have the property that the determinant of A= 0 if and only if A is singular. First we need some terminology.

5. Minors and cofactors If A is a square matrix, then the minor M ij of the element a ij of A is the determinant of the matrix obtained by deleting the ith row and the jth column from A. The cofactor C ij = (-1) i+j M ij.

6. Definition of a Determinant If A is a square matrix of order 2 or greater, then the determinant of A is the sum of the entries in the first row of A multiplied by their cofactors. That is,

7. LaPlace’s expansion of a determinant Theorem: Let A be a square matrix of order n. Then for any i,j, or