Determinant of a square matrix

L. El Ghaoui

Determinant of a square matrix

Definition

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

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

where $A_{i,1}$ is the $(n-1) \times (n-1)$ matrix obtained from $A$ by removing the $i$ -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 $A$ such that

$\det I = 1$ .
$A \rightarrow \det A$ is a linear function of any column (when the others are fixed).
$\det A$ changes sign when two columns are permuted.

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

$\det A = \sum\limits_{\sigma \in S_n} {\bf sign}(\sigma) A_{\sigma(i), i}$

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

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 $A$ with columns $r_1, r_2, r_3$ is the volume of the parallelepiped defined by the vectors $r_1, r_2, r_3$ . (Source: wikipedia). Hence the determinant is a measure of scale that quantifies how the linear map associated with $A, x \rightarrow Ax$ , changes volumes.

In general, the absolute value of the determinant of a $n \times n$ matrix is the volume of the parallelepiped

$\{Ax: 0\leq x_i \leq 1, \quad i = 1, \dots, n\}.$

This is consistent with the fact that when $A$ 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 $A$ : 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 $A$ by removing its -th row and -th column. For example, the determinant of a $2 \times 2$ matrix

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

is given by

$\det A = ad-bc.$

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

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

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.