Section 6.5 Determinant
This section is a brief review of determinants from linear algebra in preparation for the change of variables formula of integrals. The determinant is a scalar valued function \(A\mapsto \det A\) on the space of square matrices with the following properties:
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\(\det I_n=1\) for any \(n\times n\) identity matrix \(I_n\text{.}\)
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\(A\mapsto \det A\) is a linear function of each of the column vectors of \(A\) when the other columns are held fixed.
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If \(B\) is obtained from \(A\) by interchanging two columns, then\begin{equation*} \det B=-\det A\text{.} \end{equation*}
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If \(A\) has two equal columns, then \(\det A=0\text{.}\)
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If \(B\) is obtained from \(A\) by adding a multiple of one column of \(A\) to a different column, then \(\det B=\det A\text{.}\)
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\(\det (AB)=\det A \det B\) for any two \(n\times n\) matrices \(A, B\text{.}\)
In fact properties (d)-(f) follow from (a)-(c).
One can either produce a formula for \(\det A\) in terms of the entries of \(A\) and verify that it satisfies the above properties, or use these properties to prove such a value of the determinant is uniquely determined---this will give an algorithm for computing \(\det A\) based on the above properties. It turns out that the above properties indeed determine the \(\det A\) uniquely. Its formula can be found in any standard textbook on linear algebra, but is rarely used directly except for \(2\times 2\) matrices.
These properties are rooted in the geometric origin in the determinant function.
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Properties (a)-(e) encapsulate the properties of signed area (volume) of parallelograms (parallelepipeds) with edges formed using the column vectors of the matrix: property (e) is a reflection of the geometric property that parallelograms with equal base and equal heights have equal areas-the above operation corresponds to fixing an \((n-1)\) dimensional base and sliding the edge not in the base in the direction of a base edge, thus resulting in a newly formed parallelogram (parallelepiped) with the same base but equal height.It may be appealing to have a formula to directly compute the geometric area (volume) of a parallelogram (parallelepiped). But such a formula would lose the linearity as in (b). We would rather keep (b), and this necessitates in allowing negative values in the determinant, so the geometric interpretation has to be signed area (volume), which is related to the orientation of the edges of the parallelogram (parallelepiped)-for a \(2\times 2\) matrix, it reflects whether the relation between the first and second columns is counter-clockwise or clockwise rotation.We can still use \(\vert \det A \vert \) to represent the geometric area (volume) of a parallelogram (parallelepiped). It fails the linearity but still maintains the following three geometric properties.
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If a multiple of one column of \(A\) is added to another column to form \(B\text{,}\) then \(|\det B|=|\det A|\text{.}\)
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If one column of \(A\) is a linear combination of the rest of the columns of \(A\text{,}\) then \(|\det A|=0\text{.}\) The converse also holds.
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If one column of \(A\) is multiplied by a scalar \(c\) to form a matrix \(B\text{,}\) then \(|\det B|=|c| |\det A|\text{.}\)
An equivalent way of stating property (ii) is that \(\det A=0\) iff the system \(A\bx =\mathbf 0\) has a non-zero solution. -
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Any linear map associated to a square matrix \(A\) maps a region \(D\) to its image \(A(D)\text{.}\) Although we have not discussed the notion of area (volume) of a general region, the notion is intuitively clear at least when \(D\) is the non-overlapping union of rectangles (the interior of the rectangles are not allowed to overlap, but the edges are allowed to overlap). Then the ratio of the areas (volumes) of \(A(D)\) and \(D\) is independent of \(D\text{,}\) and equals \(|\det A|\text{.}\) In other words, \(|\det A|\) is the magnifying factor of area (volume) for \(A\) as a map. The property \(\det (BA)=\det B \det A\) is a reflection of this perspective.
The following is a Desmos graph illustration of areas of parallelograms with adjacent edges [u, v] and [u, v+pu].
