Matrices and Determinants
The determinant of a matrix
Expanding a determinant by cofactors
Calculating the value of a determinant
The determinant of a 2-by-2 matrix
The determinant of a 3-by-3 matrix
The determinant of a matrix
Determinants are defined only for square matrices.
The determinant D of an n-by-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.
Expanding a determinant by cofactors
That is, the value of a determinant equals the sum of the products of the entries in anyone row or column and their respective cofactors.
A cofactor of an entry aik is a signed minor or (a sub-determinant) derived from a given matrix or determinant by the deletion of the row and column containing the specified entry.
Therefore, the cofactor of the entry aik of a matrix A
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.
A determinant can be expanded along any row or column without affecting the determinant's value.
Thus, the value of a determinant of order n expanded along the kth column
where k can be any column between 1 and n.
Calculating the value of a determinant
The determinant of a 2-by-2 matrix
Let apply the above formula to calculate the value of the determinant of a 2 ´ 2 matrix A,
then, the expansion of the above determinant by the first column
thus, obtained is
The determinant of a 3-by-3 matrix
Using the above determinant expansion by cofactors formula we calculate the value of the determinant of a     3 ´ 3 matrix A,
The diagram below shows the method of determining cofactors of the 2nd column's entries, as we chose to expand the above determinant along that column
thus, obtained is
Example:  For the given 3-by-3 matrix A, find the value of the determinant D = det(A),
By expanding the determinant by the first column obtained is
Matrices, determinants and systems of linear equations contents