Determinant of a square matrix

Definition
An important result
Geometry
Formula
Determinant and inverse
Some properties

We review a few important facts abut determinants here. For a self-contained exposition that includes proofs, see this text by Carl de Boor.

Definition

The determinant of a square, n times n matrix , denoted det A , is defined by an algebraic formula of the coefficients of . The following formula for the determinant, known as Laplace's expansion formula, allows to compute the determinant recursively:

$det A = sum_{i=1}^n (-1)^{i+1} A_{i,1}det (C_i),$

where $C_{i,1}$ is the (n-1) times (n-1) matrix obtained from by removing the -th row and first column. (The first column does not play a special role here: the determinant remains the same if we use any other column.)

The determinant is the unique function of the entries of such that

.
is a linear function of any column (when the others are fixed).
changes sign when two columns are permuted.

There are other expressions of the determinant, including Leibnitz formula (proven here):

$det A = sum_{sigma in S_n} : mbox{bf sign}(sigma) A_{sigma(i),i}$

where S_n denotes the set of permutations sigma of the integers 1,2,ldots,n . Here, mbox{bf sign}(sigma) denotes the sign of the permutation sigma , which is the number of pairwise exchanges required to transform sigma(1),sigma(2),ldots,sigma(n) into .

Important result

An important result is that a square matrix is invertible if and only if its determinant is not zero. We use this key result when introducing eigenvalues of symmetric matrices.

Geometry

The determinant of a 3 times 3 matrix with columns r_1,r_2,r_3 is the volume of the parallepiped defined by the vectors . (Source: wikipedia). Hence the determinant is a measure of scale that quantifies how the linear map associated with , x rightarrow Ax , changes volumes.

In general, the absolute value of the determinant of a n times n matrix is the volume of the parallepiped

$left{ Ax ~:~ 0 le x_i le 1, ;; i=1,ldots,n right}.$

This is consistent with the fact that when is not invertible, its columns define a parallepiped of zero volume.

Determinant and inverse

The determinant can be used to compute the inverse of a square, full rank (that is, invertible) matrix : the inverse $B = A^{-1}$ has elements given by

$B_{ij} = frac{(-1)^{i+j}}{det A} det (tilde{A}_{ij}),$

, where $tilde{A}_{ij}$ is a matrix obtained from

by removing its

-th row and

-th column. For example, the determinant of a 2 times 2

matrix

A = left( begin{array}{cc} a & b c & d end{array}right)

is given by

It is indeed the volume of the area of a parallepiped defined with the columns of , (a,c),(b,d) . The inverse is given by

$A^{-1} = frac{1}{(ad-bc)} left( begin{array}{cc} d & -b -c & a end{array}right)$

Some properties

Determinant of triangular matrices

If a matrix is square, triangular, then its determinant is simply the product of its diagonal coefficients. This comes right from Laplace's expansion formula above.