Determinant of a square matrix
Definition
The determinant of a square, matrix , denoted , 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:
where is the 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
1. .
2. is a linear function of any column (when the others are fixed).
3. changes sign when two columns are permuted.
There are other expressions of the determinant, including the Leibnitz formula (proven here):
where denotes the set of permutations of the integers . Here, denotes the sign of the permutation , which is the number of pairwise exchanges required to transform 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 matrix with columns is the volume of the parallelepiped defined by the vectors . (Source: wikipedia). Hence the determinant is a measure of scale that quantifies how the linear map associated with , changes volumes. |
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In general, the absolute value of the determinant of a matrix is the volume of the parallelepiped
This is consistent with the fact that when is not invertible, its columns define a parallelepiped 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 has elements given by
where is a matrix obtained from by removing its -th row and -th column. For example, the determinant of a matrix
is given by
It is indeed the volume of the area of a parallelepiped defined with the columns of , . The inverse is given by
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.
Determinant of transpose
The determinant of a square matrix and that of its transpose are equal.
Determinant of a product of matrices
For two invertible square matrices, we have
In particular:
This also implies that for an orthogonal matrix , that is, a matrix with , we have
Determinant of block matrices
As a generalization of the above result, we have three compatible blocks :
A more general formula is