Definition

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

[latex]\begin{align*} \det A &= \sum\limits_{i=1}^n (-1)^{i+1} A_{i,1} \det(C_i), \end{align*}[/latex]

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

1. [latex]\det I = 1[/latex].

2. [latex]A \rightarrow \det A[/latex] is a linear function of any column (when the others are fixed).

3. [latex]\det A[/latex] changes sign when two columns are permuted.

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

[latex]\begin{align*} \det A &= \sum\limits_{\sigma \in S_n} {\bf sign}(\sigma) A_{\sigma(i), i} \end{align*}[/latex]

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

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

In general, the absolute value of the determinant of a [latex]n \times n[/latex] matrix is the volume of the parallelepiped

[latex]\begin{align*} \{Ax: 0\leq x_i \leq 1, \quad i = 1, \dots, n\}. \end{align*}[/latex]

This is consistent with the fact that when [latex]A[/latex] 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 [latex]A[/latex]: the inverse [latex]B=A^{-1}[/latex] has elements given by

[latex]\begin{align*} B_{ij} &= \frac{(-1)^{i+j}}{\det A} \det (\tilde{A}_{ij}), \end{align*}[/latex]

where [latex]\tilde{A}_{ij}[/latex] is a matrix obtained from [latex]A[/latex] by removing its -th row and -th column. For example, the determinant of a [latex]2 \times 2[/latex] matrix

[latex]\begin{align*} A &= \left(\begin{array}{ll} a & b \\ c & d \end{array}\right) \end{align*}[/latex]

is given by

[latex]\begin{align*} \det A &= ad-bc. \end{align*}[/latex]

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

[latex]\begin{align*} A^{-1} &= \frac{1}{ad-bc} \left(\begin{array}{cc} \phantom{-}d & -b \\ -c & \phantom{-}a \end{array}\right) \end{align*} [/latex]

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

[latex]\begin{align*} \det AB &= \det A \cdot \det B. \end{align*}[/latex]

In particular:

[latex]\begin{align*} \det A^{-1} &= \frac{1}{\det A}. \end{align*}[/latex]

This also implies that for an orthogonal matrix [latex]U[/latex], that is, a [latex]n \times n[/latex] matrix with [latex]U^TU = I[/latex], we have

[latex]\begin{align*} 1 &= \det U^TU = (\det U^T) \det U = (\det U)^2. \end{align*}[/latex]

Determinant of block matrices

As a generalization of the above result, we have three compatible blocks [latex]A, C, D[/latex]:

A more general formula is