Why is the determinant the volume of a parallelepiped in any dimensions?

Question

For $n = 2$, I can visualize that the determinant $n \times n$ matrix is the area of the parallelograms by actually calculating the area by coordinates. But how can one easily realize that it is true for any dimensions?

Answer 1

If the column vectors are linearly dependent, both the determinant and the volume are zero. So assume linear independence. The determinant remains unchanged when adding multiples of one column to another. This corresponds to a skew translation of the parallelepiped, which does not affect its volume. By a finite sequence of such operations, you can transform your matrix to diagonal form, where the relation between determinant (=product of diagonal entries) and volume of a "rectangle" (=product of side lengths) is apparent.

Answer 2

Here is the same argument as Muphrid's, perhaps written in an elementary way.

Apply Gram-Schmidt orthogonalization to $\{v_{1},\ldots,v_{n}\}$, so that \begin{eqnarray*} v_{1} & = & v_{1}\\ v_{2} & = & c_{12}v_{1}+v_{2}^{\perp}\\ v_{3} & = & c_{13}v_{1}+c_{23}v_{2}+v_{3}^{\perp}\\ & \vdots \end{eqnarray*} where $v_{2}^{\perp}$ is orthogonal to $v_{1}$; and $v_{3}^{\perp}$ is orthogonal to $span\left\{ v_{1},v_{2}\right\} $, etc.

Since determinant is multilinear, anti-symmetric, then \begin{eqnarray*} \det\left(v_{1},v_{2},v_{3},\ldots,v_{n}\right) & = & \det\left(v_{1},c_{12}v_{1}+v_{2}^{\perp},c_{13}v_{1}+c_{23}v_{2}+v_{3}^{\perp},\ldots\right)\\ & = & \det\left(v_{1},v_{2}^{\perp},v_{3}^{\perp},\ldots,v_{n}^{\perp}\right)\\ & = & \mbox{signed volume}\left(v_{1},\ldots,v_{n}\right) \end{eqnarray*}

Answer 3

In 2d, you calculate the area of a parallelogram spanned by two vectors using the cross product. In 3d, you calculate the volume of a parallelepiped using the triple scalar product. Both of these can be written in terms of a determinant, but it's probably not clear to you what the proper generalization is to higher dimensions.

That generalization is called the wedge product. Given $n$ vectors $v_1, v_2, \ldots, v_n$, the wedge product $v_1 \wedge v_2 \wedge \ldots \wedge v_n$ is called an $n$-vector, and it has as its magnitude the $n$-volume of that $n$-parallelepiped.

What is the relationship between the wedge product and the determinant? Quite simple, actually. There is a natural generalization of linear maps to work on $k$-vectors. Given a linear map $\underline T$ (which can be represented as a matrix), the action of that map on a $k$-vector is defined as

$$\underline T(v_1 \wedge v_2 \wedge \ldots \wedge v_k) \equiv \underline T(v_1) \wedge \underline T(v_2) \wedge \ldots \wedge \underline T(v_k)$$

When talking about $n$-vectors in an $n$-dimensional space, it's important to realize that the "vector space" of these $n$-vectors is in fact one-dimensional. That is, if you think about volume, there is only one such unit volume in a given space, and all other volumes are just scalar multiples of it. Hence, when we talk about the action of a linear map on an $n$-vector, we can see that

$$\underline T(v_1 \wedge v_2 \wedge \ldots \wedge v_n) = \alpha [v_1 \wedge v_2 \wedge \ldots \wedge v_n]$$

for some scalar $\alpha$. In fact, this is a coordinate system independent definition of the determinant!

When you build a matrix out of $n$ vectors $f_1, f_2, \ldots, f_n$ as the matrix's columns, what you're really doing is the following: you're saying that, if you have a basis $e_1, e_2, \ldots, e_n$, then you're defining a map $\underline T$ such that $\underline T(e_1) = f_1$, $\underline T(e_2) = f_2$, and so on. So when you input $e_1 \wedge e_2 \wedge \ldots \wedge e_n$, you get

$$\underline T(e_1 \wedge e_2 \wedge \ldots \wedge e_n) = (\det \underline T) e_1 \wedge e_2 \wedge \ldots \wedge e_n= f_1 \wedge f_2 \wedge \ldots \wedge f_n$$

This is how you can use a matrix determinant to calculate volumes: it's just an easy way of constructing something that automatically computes the wedge product.

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Edit: how one can see that the wedge product accurately gives the volume of a parallelepiped. Any vector can be broken down into perpendicular and parallel parts with respect to another vector, to a plane, and so on (or to any $k$-vector). As such, if I have two vectors $a$ and $b$, then the wedge product $a \wedge b = a \wedge b_\perp$, where $b_\perp$ is effectively the height of the parallelogram. Similarly, if I construct a parallelepiped with a vector $c$, then the wedge product $a \wedge b \wedge c = (a \wedge b_\perp) \wedge c_\perp$, where $c_\perp$ lies entirely normal to $a \wedge b_\perp$. So we can recursively do this for any $k$-vector, looking at orthogonal vectors instead, which is much simpler to see the volumes from.

Answer 4

The determinant of a matrix A is the unique function that satisfies:

1. $\det(A)=0$ when two columns are equal
2. the determinant is linear in the columns
3. if A is the identity $\det(A)=1$.

You can easily convince yourself that the oriented volume $\operatorname{vol}(v_1,v_2,\ldots,v_n)$ between $v_1, v_2,\ldots, v_n$ vectors is a function that satisfies exactly the same properties if we place the vectors as the columns of a matrix $A=(v_1,\ldots,v_n)$. Hence $\operatorname{vol}(v_1,v_2,\ldots,v_n)=\det(A)$.

Answer 5

Determinant involves a cross-product of the first two vectors and a dot of the result with the third. The result of a cross product is a vector whose magnitude is the area of its null space. Said simply, any plane in 3D is the null space of its normal.The size of the plane is defined by the length of the normal. The volume is found by projecting this normal onto the third vector.

Answer 6

You can also invoke the change-of-variable theorem in higher dimensions. An $n$-dimensional parallelepiped $\mathcal P=\mathcal P(a_1,\dots,a_n)$ in $\mathbb R^n$ (where the $a_i$ are independent vectors in $\mathbb R^n$) is the set of all $x$ such that: $$ x=c_1a_1+\dots+c_ka_n, $$ with $0\leq c_i\leq 1$. We can define the linear transformation $h(x)=A\cdot x$, where $A$ is the $n\times n$ matrix with $a_i$ as its columns. This gives us $\mathcal P=h([0,1]^n)$. The volume of $h([0,1]^n)$ is equal to $h((0,1)^n))$ (those sets are equal modulo a set of measure zero), so we can apply the change-of-variable theorem: $$ v(\mathcal P)=\int_{h((0,1)^n)}1=\int_{(0,1)^n}\vert\det Dh\vert=\vert\det A\vert. $$

Edit:

As pointed out in the comments, conceptually this argument is circular. Alternatively, one can give a proof that the $n$-dimensional Lebesgue measure $\lambda^n$ transforms in this way, i.e.:

Given an (invertible) linear map $A:\mathbb R^n\to\mathbb R^n$, we have $$ \lambda^n(A(B))=\vert\det A\vert\lambda^n(B) $$ for any Borel measurable set $B$ of $\mathbb R^n$.

The proof uses the push-forward measure $\lambda^n(A(-))$ and invokes the uniqueness${}^{(1)}$ of the Lebesgue measure, along with some elementary linear algebra: first show that the transformation rule holds for diagonal matrices and orthogonal matrices, and then use that any invertible matrix $A$ can be written as $UDV$ where $U,V$ are orthogonal and $D$ is diagonal.

${}^{(1)}$ For any measure $\mu$ on $\mathbb R^n$ such that $\mu([0,1]^n)<\infty$ we have $\mu=\mu([0,1]^n)\cdot\lambda^n$.

For the details see Schilling's book Measures, Integrals and Martingales (Appendix C The Volume of a Parallelepiped).