Determinant

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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. If the characteristic polynomial splits into linear factors, then he determinant is equal to the product of the eigenvalues of a matrix, counted by their algebraic multiplicities.

Contents

Motivation

A matrix can be used to transform a geometric figure. For example, in the plane, if we have a triangle defined by its vertices (3,3), (5,1), and (1,4), and we wish to transform this triangle into the triangle of vertices (3,-3), (5,-9), and (1,2), we can simply do a matrix multiplication of each vertex by the matrix \begin{pmatrix}
1  & 0 \\
-2 & 1 \\
\end{pmatrix}.

In this transformation, no matter what is the shape of the initial geometric figure, its position, or its area, the final geometric figure will have the same area and orientation.

It can be seen that matrix transformations of geometric figures always give resulting figures whose area is proportional to the initial figure, and whose orientation is either always the same, or always the reverse.

This ratio is called the determinant of the matrix, and it's positive when the orienation is kept, negative when the orientation is reversed, and zero when the final figure always has zero area.

This two-dimensional concept is easily generalized for any dimensions. In 3D, replace area for volume, and in higher dimensions the analogue concept is called hypervolume.

The determinant of a matrix is the oriented ratio of the hypervolumes of the transformed figure to the source figure.

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 mij of the element aij 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 Cij equals the minor mij 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 | = | MT | 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 α 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).

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