Editing Determinant (section)

==Geometric meaning==
[[File:Area parallelogram as determinant modified.svg|thumb|right|The area of the parallelogram is the absolute value of the determinant of the matrix formed by the vectors representing the parallelogram's sides.]]

If the matrix entries are real numbers, the matrix {{math|A}} represents the [[linear map]] that maps the [[basis vector]]s to the columns of {{math|A}}.  The images of the basis vectors form a [[parallelogram]] that represents the image of the [[unit square]] under the mapping.  The parallelogram defined by the columns of the above matrix is the one with vertices at {{math|(0, 0)}}, {{math|(''a'', ''c'')}}, {{math|(''a'' + ''b'', ''c'' + ''d'')}}, and {{math|(''b'', ''d'')}}, as shown in the accompanying diagram.

The absolute value of {{math|''ad'' − ''bc''}} is the area of the parallelogram, and thus represents the scale factor by which areas are transformed by {{math|A}}.

The absolute value of the determinant together with the sign becomes the [[signed area]] of the parallelogram. The signed area is the same as the usual [[area (geometry)|area]], except that it is negative when the angle from the first to the second vector defining the parallelogram turns in a clockwise direction (which is opposite to the direction one would get for the [[identity matrix]]).

To show that {{math|''ad'' − ''bc''}} is the signed area, one may consider a matrix containing two vectors {{math|'''u''' ≡ (''a'', ''c'')}} and {{math|'''v''' ≡ (''b'', ''d'')}} representing the parallelogram's sides. The signed area can be expressed as {{math|{{!}}'''u'''{{!}} {{!}}'''v'''{{!}} sin ''θ''}} for the angle ''θ'' between the vectors, which is simply base times height, the length of one vector times the perpendicular component of the other. Due to the [[sine]] this already is the signed area, yet it may be expressed more conveniently using the [[cosine]] of the complementary angle to a perpendicular vector, e.g. {{math|1='''u'''<sup>⊥</sup> = (−''c'', ''a'')}}, so that {{math|{{!}}'''u'''<sup>⊥</sup>{{!}} {{!}}'''v'''{{!}} cos ''θ&prime;''}} becomes the signed area in question, which can be determined by the pattern of the [[scalar product]] to be equal to {{math|''ad'' − ''bc''}} according to the following equations:

: <math>\text{Signed area} =
  |\boldsymbol{u}|\,|\boldsymbol{v}|\,\sin\,\theta = \left|\boldsymbol{u}^\perp\right|\,\left|\boldsymbol{v}\right|\,\cos\,\theta' =
  \begin{pmatrix} -c \\ a \end{pmatrix} \cdot \begin{pmatrix} b \\ d \end{pmatrix} = ad - bc.
</math>

Thus the determinant gives the area scale factor and the orientation induced by the mapping represented by ''A''. When the determinant is equal to one, the linear mapping defined by the matrix preserves area and orientation.

[[File:Determinant parallelepiped.svg|300px|right|thumb|The volume of this [[parallelepiped]] is the absolute value of the determinant of the matrix formed by the columns constructed from the vectors r1, r2, and r3.]]
If an {{math|''n'' × ''n''}} [[Real number|real]] matrix ''A'' is written in terms of its column vectors <math>A = \left[\begin{array}{c|c|c|c} \mathbf{a}_1 & \mathbf{a}_2 & \cdots & \mathbf{a}_n\end{array}\right]</math>, then
:<math>
  A\begin{pmatrix}1 \\ 0\\ \vdots \\0\end{pmatrix} = \mathbf{a}_1, \quad
  A\begin{pmatrix}0 \\ 1\\ \vdots \\0\end{pmatrix} = \mathbf{a}_2, \quad
                                                           \ldots, \quad
  A\begin{pmatrix}0 \\0 \\ \vdots \\1\end{pmatrix} = \mathbf{a}_n.
</math>

This means that <math>A</math> maps the unit [[Hypercube|''n''-cube]] to the ''n''-dimensional [[parallelepiped#Parallelotope|parallelotope]] defined by the vectors <math>\mathbf{a}_1, \mathbf{a}_2, \ldots, \mathbf{a}_n,</math> the region <math>P = \left\{c_1 \mathbf{a}_1 + \cdots + c_n\mathbf{a}_n \mid 0 \leq c_i\leq 1 \ \forall i\right\}</math> (<math display="inline">\forall</math> stands for "for all" as a [[List of logic symbols|logical symbol]].)

The determinant gives the [[orientation (vector space)|signed]] ''n''-dimensional volume of this parallelotope, <math>\det(A) = \pm \text{vol}(P),</math> and hence describes more generally the ''n''-dimensional volume scale factor of the [[linear transformation]] produced by ''A''.<ref>{{cite web|url=https://textbooks.math.gatech.edu/ila/determinants-volumes.html|title=Determinants and Volumes|website=textbooks.math.gatech.edu|access-date=16 March 2018}}</ref> (The sign shows whether the transformation preserves or reverses [[Orientation (vector space)|orientation]].) In particular, if the determinant is zero, then this parallelotope has volume zero and is not fully ''n''-dimensional, which indicates that the dimension of the image of ''A'' is less than ''n''. This [[Rank–nullity theorem|means]] that ''A'' produces a linear transformation which is neither [[surjective function|onto]] nor [[Injective function|one-to-one]], and so is not invertible.