Matrices and Determinants The determinant of a matrix
Properties of determinants The inverse of a matrix Method of expanding a determinant of a rank n by cofactors
The determinant of a matrix
Determinants are defined only for square matrices.
The determinant D of an n ´ n matrix A, also denoted det(A) or |A| is an ordered square array of elements the value of which is given by an alternating sum of products of the elements of A, that can be obtained by using the method of expanding the determinant to cofactors.
Properties of determinants
Use of the following properties simplify calculation of the value of higher order determinants.
1. Interchanging (switching) two rows or columns changes the sign of the determinant.
2. A determinant with a row or column of zeros has value 0.
3. A determinant with two rows or columns equal (or proportional) has value 0.
4. A common factor of all elements of a row or a column can be factored out.
5. The multiplication of a determinant by a scalar can be done by multiplying each element of any row or column by the scalar.
6. The determinant of a triangular matrix is equal to the product of the diagonal elements.
7. The value of a determinant will not change by adding multiples of any column or row to any other column or row.
8. A determinant can be expanded along any row or column (without affecting the determinant's value).
Therefore, we use the above properties to set as many zeros in a row or column as possible to simplify calculation of the value of a determinant.
Example:  Let find the value of the determinant of the matrix A from the previous example applying the method of expanding the determinant to cofactors and using properties of determinants. First, added is the second row to the first writing the sum to the first row. Then we add third row to second, writing the sum to the second row. Finally, the first column multiplied by -1 we added to the second column to obtain the triangular matrix.
By expanding the above triangular matrix by the first column obtained is This way proved is the 6th property stating that the determinant of a triangular matrix is equal to the product of its diagonal elements.
The inverse of a matrix
The inverse matrix A-1, of a square matrix A, is the matrix which when multiplied by A on either side gives the identity matrix I (the identity matrix is a square matrix in which every main diagonal entry is 1 and the other elements are all zero),
AA-1 = A-1A = I .
The inverse of a regular square matrix A = [aik] (with a non zero determinant) we obtain dividing each entry of the transpose of the cofactor matrix of A, called the adjoint matrix, by the value of determinant of A, i.e., The entries of the cofactor matrix [Aik]
Aik = (-1)i + k · Dik
where Dik is the minor (or subdeterminant) of the matrix A obtained by deleting its ith row and kth column.
Interchange the rows and columns of the cofactor matrix to obtain [Aki] the transpose of the cofactor matrix.
Example:  Find the inverse of the 3 ´ 3 matrix A from the previous example.
Solution:  By expanding the det(A) by the second row obtained is, Then calculate the cofactors Aik, to write the cofactor matrix [Aik] By flipping the cofactor matrix [Aik] around the main diagonal obtained is the adjoint matrix [Aki] or the transpose of the cofactor matrix.
Therefore, the inverse of the matrix A Let verify that the matrix A multiplied by its inverse gives the identity matrix,    Matrices, determinants and systems of linear equations contents 