Determinant

The determinant of a matrix (written |A|) is a single number that depends on the elements of the matrix A. Determinants exist only for square matrices (i.e., ones where the number of rows equals the number of columns).

Determinants are a basic building block of linear algebra, and are useful for finding areas and volumes of geometric figures, in Cramer's Rule, and in many other areas ways.

How to calculate

We need to introduce two notions: the minor and the cofactor of a matrix element. Also, the determinant of a 1x1 matrix equals the sole element of that matrix.

Minor

The minor $m_{ij}$ of the element $a_{ij}$ of an NxN matrix is the determinant of the (N-1)x(N-1) matrix formed by removing the ith row and jth column from M.

Cofactor

The cofactor $C_{ij}$ equals the minor $m_{ij}$ multiplied by $(-1)^{i+j}$

The determinant is then defined to be the sum of the products of the elements of any one row or column with their corresponding cofactors.

2x2 case

For the 2x2 matrix

${\begin{pmatrix}a&b\\c&d\end{pmatrix}}$

the determinant is simply ad-bc (for example, using the above rule on the first row).

3x3 case

For a general 3x3 matrix

${\begin{pmatrix}A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&A_{33}\\\end{pmatrix}}$

we can expand along the first row to find

$|A|=A_{11}{\begin{vmatrix}A_{22}&A_{23}\\A_{32}&A_{33}\end{vmatrix}}-A_{12}{\begin{vmatrix}A_{21}&A_{23}\\A_{31}&A_{33}\end{vmatrix}}+A_{13}{\begin{vmatrix}A_{21}&A_{22}\\A_{31}&A_{32}\end{vmatrix}}$

where each of the 2x2 determinants is given above.

Properties of determinants

The following are some useful properties of determinants. Some are useful computational aids for simplifying the algebra needed to calculate a determinant. The first property is that $|M|=|M^{T}|$ where the superscript "T" denotes transposition. Thus, although the following rules refer to the rows of a matrix they apply equally well to the columns.

• The determinant is unchanged by adding a multiple of one row to any other row.
• If two rows are interchanged the sign of the determinant will change
• If a common factor $\alpha$ is factored out from each element of a single row, the determinant is multiplied by that same factor.
• If all the elements of a single row are zero (or can be made to be zero using the above rules) then the determinant is zero.
• $|AB|=|A||B|$

In practice, one of the most efficient ways of finding the determinant of a large matrix is to add multiples of rows and/or columns until the matrix is in triangular form such that all the elements above or below the diagonal are zero, for example

${\begin{pmatrix}A_{11}&A_{12}&A_{13}\\0&A_{22}&A_{23}\\0&0&A_{33}\\\end{pmatrix}}$.

The determinant of such a matrix is simply the product of the diagonal elements (use the cofactor expansion discussed above and expand down the first column).