Editing Determinant (section)

== Applications ==

=== Cramer's rule ===

Determinants can be used to describe the solutions of a [[linear system of equations]], written in matrix form as <math>Ax = b</math>. This equation has a unique solution <math>x</math> if and only if <math>\det (A)</math> is nonzero. In this case, the solution is given by [[Cramer's rule]]:
:<math>x_i = \frac{\det(A_i)}{\det(A)} \qquad i = 1, 2, 3, \ldots, n</math>

where <math>A_i</math> is the matrix formed by replacing the <math>i</math>-th column of <math>A</math> by the column vector <math>b</math>. This follows immediately by column expansion of the determinant, i.e.
:<math>\det(A_i) =
  \det\begin{bmatrix}a_1 & \ldots & b & \ldots & a_n\end{bmatrix} 
</math>
<math>
  =\sum_{j=1}^n x_j\det\begin{bmatrix}a_1 & \ldots & a_{i-1} & a_j & a_{i+1} & \ldots & a_n\end{bmatrix} =
  x_i\det(A)
</math>

where the vectors <math>a_j</math> are the columns of ''A''.  The rule is also implied by the identity

:<math>A\, \operatorname{adj}(A) = \operatorname{adj}(A)\, A = \det(A)\, I_n.</math>

Cramer's rule can be implemented in <math>\operatorname O(n^3)</math> time, which is comparable to more common methods of solving systems of linear equations, such as [[LU decomposition|LU]], [[QR decomposition|QR]], or [[singular value decomposition]].<ref>{{harvnb|Habgood|Arel|2012}}</ref>

=== Linear independence ===
Determinants can be used to characterize [[linear independence|linearly dependent]] vectors: <math>\det A</math> is zero if and only if the column vectors of the matrix <math>A</math> are linearly dependent.<ref>{{harvnb|Lang|1985|loc=§VII.3}}</ref> For example, given two linearly independent vectors <math>v_1, v_2 \in \mathbf R^3</math>, a third vector <math>v_3</math> lies in the [[Plane (geometry)|plane]] [[Linear span|spanned]] by the former two vectors exactly if the determinant of the <math>3 \times 3</math> matrix consisting of the three vectors is zero. The same idea is also used in the theory of [[differential equation]]s: given functions <math>f_1(x), \dots, f_n(x)</math> (supposed to be <math>n-1</math> times [[differentiable function|differentiable]]), the [[Wronskian]] is defined to be
:<math>W(f_1, \ldots, f_n)(x) =
  \begin{vmatrix}
            f_1(x) &         f_2(x) & \cdots &         f_n(x) \\
           f_1'(x) &        f_2'(x) & \cdots &        f_n'(x) \\
            \vdots &         \vdots & \ddots &         \vdots \\
    f_1^{(n-1)}(x) & f_2^{(n-1)}(x) & \cdots & f_n^{(n-1)}(x)
\end{vmatrix}.</math>

It is non-zero (for some <math>x</math>) in a specified interval if and only if the given functions and all their derivatives up to order <math>n-1</math> are linearly independent. If it can be shown that the Wronskian is zero everywhere on an interval then, in the case of [[analytic function]]s, this implies the given functions are linearly dependent. See [[Wronskian#The Wronskian and linear independence|the Wronskian and linear independence]]. Another such use of the determinant is the [[resultant]], which gives a criterion when two [[polynomial]]s have a common [[root of a function|root]].<ref>{{harvnb|Lang|2002|loc=§IV.8}}</ref>

=== Orientation of a basis ===
{{Main|Orientation (vector space)}}
The determinant can be thought of as assigning a number to every [[sequence]] of ''n'' vectors in '''R'''<sup>''n''</sup>, by using the square matrix whose columns are the given vectors. The determinant will be nonzero if and only if the sequence of vectors is a ''basis'' for '''R'''<sup>''n''</sup>. In that case, the sign of the determinant determines whether the [[orientation (vector space)|orientation]] of the basis is consistent with or opposite to the orientation of the [[standard basis]]. In the case of an orthogonal basis, the magnitude of the determinant is equal to the ''product'' of the lengths of the basis vectors.  For instance, an [[orthogonal matrix]] with entries in '''R'''<sup>''n''</sup> represents an [[orthonormal basis]] in [[Euclidean space]], and hence has determinant of ±1 (since all the vectors have length 1). The determinant is +1 if and only if the basis has the same orientation. It is −1 if and only if the basis has the opposite orientation.

More generally, if the determinant of ''A'' is positive, ''A'' represents an orientation-preserving [[linear transformation]] (if ''A'' is an orthogonal {{math|2 × 2}} or {{math|3 × 3}} matrix, this is a [[rotation (mathematics)|rotation]]), while if it is negative, ''A'' switches the orientation of the basis.

=== Volume and Jacobian determinant ===
As pointed out above, the [[absolute value]] of the determinant of real vectors is equal to the volume of the [[parallelepiped]] spanned by those vectors. As a consequence, if <math>f : \mathbf R^n \to \mathbf R^n</math> is the linear map given by multiplication with a matrix <math>A</math>, and <math>S \subset \mathbf R^n</math> is any [[Lebesgue measure|measurable]] [[subset]], then the volume of <math>f(S)</math> is given by <math>|\det(A)|</math> times the volume of <math>S</math>.<ref>{{harvnb|Lang|1985|loc=§VII.6, Theorem 6.10}}</ref> More generally, if the linear map <math>f : \mathbf R^n \to \mathbf R^m</math> is represented by the <math>m \times n</math> matrix <math>A</math>, then the <math>n</math>-[[dimension]]al volume of <math>f(S)</math> is given by:
:<math>\operatorname{volume}(f(S)) = \sqrt{\det\left(A^\textsf{T} A\right)} \operatorname{volume}(S).</math>

By calculating the volume of the [[tetrahedron]] bounded by four points, they can be used to identify [[skew line]]s. The volume of any tetrahedron, given its [[vertex (geometry)|vertices]] <math>a, b, c, d</math>, <math>\frac 1 6 \cdot |\det(a-b,b-c,c-d)|</math>, or any other combination of pairs of vertices that form a [[spanning tree]] over the vertices.

[[File:Jacobian_determinant_and_distortion.svg|350px|thumb|right|A nonlinear map <math>f \colon \mathbf{R}^{2} \to \mathbf{R}^{2}</math> sends a small square (left, in red) to a distorted parallelogram (right, in red). The Jacobian at a point gives the best linear approximation of the distorted parallelogram near that point (right, in translucent white), and the Jacobian determinant gives the ratio of the area of the approximating parallelogram to that of the original square.]]
For a general [[differentiable function]], much of the above carries over by considering the [[Jacobian matrix]] of ''f''. For
:<math>f: \mathbf R^n \rightarrow \mathbf R^n,</math>

the Jacobian matrix is the {{math|''n'' × ''n''}} matrix whose entries are given by the [[partial derivative]]s
:<math>D(f) = \left(\frac {\partial f_i}{\partial x_j}\right)_{1 \leq i, j \leq n}.</math>

Its determinant, the [[Jacobian determinant]], appears in the higher-dimensional version of [[integration by substitution]]: for suitable functions ''f'' and an [[open subset]] ''U'' of '''R'''<sup>''n''</sup> (the domain of ''f''), the integral over ''f''(''U'') of some other function {{math|''φ'' : '''R'''<sup>''n''</sup> → '''R'''<sup>''m''</sup>}} is given by
:<math>\int_{f(U)} \phi(\mathbf{v})\, d\mathbf{v} = \int_U \phi(f(\mathbf{u})) \left|\det(\operatorname{D}f)(\mathbf{u})\right| \,d\mathbf{u}.</math>

The Jacobian also occurs in the [[inverse function theorem]].

When applied to the field of [[Cartography]], the determinant can be used to measure the rate of expansion of a map near the poles.<ref>{{Cite book|last=Lay|first=David|title=Linear Algebra and Its Applications 6th Edition|publisher=Pearson|year=2021|pages=172|language=English}}</ref>