Understand the relationship between the determinant of a matrix and the volume of a parallelepiped.

Learn to use determinants to compute volumes of parallelograms and triangles.

Learn to use determinants to compute the volume of some curvy shapes like ellipses.

Pictures: parallelepiped, the image of a curvy shape under a linear transformation.

Theorem: determinants and volumes.

Vocabulary:parallelepiped.

In this section we give a geometric interpretation of determinants, in terms of volumes. This will shed light on the reason behind three of the four defining properties of the determinant. It is also a crucial ingredient in the change-of-variables formula in multivariable calculus.

Subsection5.3.1Parallelograms and Paralellepipeds

The determinant computes the volume of the following kind of geometric object.

Definition

The paralellepiped determined by vectors in is the subset

In other words, a parallelepiped is the set of all linear combinations of vectors with coefficients in We can draw parallelepipeds using the parallelogram law for vector addition.

Example(The unit cube)

The parallelepiped determined by the standard coordinate vectors is the unit -dimensional cube.

Example(Parallelograms)

When a paralellepiped is just a paralellogram in Note that the edges come in parallel pairs.

Example

When a parallelepiped is a kind of a skewed cube. Note that the faces come in parallel pairs.

When does a parallelepiped have zero volume? This can happen only if the parallelepiped is flat, i.e., it is squashed into a lower dimension.

This means exactly that is linearly dependent, which by this corollary in Section 5.1 means that the matrix with rows has determinant zero. To summarize:

Key Observation

The parallelepiped defined by has zero volume if and only if the matrix with rows has zero determinant.

Subsection5.3.2Determinants and Volumes

The key observation above is only the beginning of the story: the volume of a parallelepiped is always a determinant.

Theorem(Determinants and volumes)

Let be vectors in let be the parallelepiped determined by these vectors, and let be the matrix with rows Then the absolute value of the determinant of is the volume of

Proof

Since the four defining properties characterize the determinant, they also characterize the absolute value of the determinant. Explicitly, is a function on square matrices which satisfies these properties:

Doing a row replacement on does not change

Scaling a row of by a scalar multiplies by

Swapping two rows of a matrix does not change

The determinant of the identity matrix is equal to

The absolute value of the determinant is the only such function: indeed, by this recipe in Section 5.1, if you do some number of row operations on to obtain a matrix in row echelon form, then

For a square matrix we abuse notation and let denote the volume of the paralellepiped determined by the rows of Then we can regard as a function from the set of square matrices to the real numbers. We will show that also satisfies the above four properties.

For simplicity, we consider a row replacement of the form The volume of a paralellepiped is the volume of its base, times its height: here the “base” is the paralellepiped determined by and the “height” is the perpendicular distance of from the base. Translating by a multiple of moves in a direction parallel to the base. This changes neither the base nor the height! Thus, is unchanged by row replacements.

For simplicity, we consider a row scale of the form This scales the length of by a factor of which also scales the perpendicular distance of from the base by a factor of Thus, is scaled by

Swapping two rows of just reorders the vectors hence has no effect on the parallelepiped determined by those vectors. Thus, is unchanged by row swaps.

The rows of the identity matrix are the standard coordinate vectors The associated paralellepiped is the unit cube, which has volume Thus,

Since is the only function satisfying these properties, we have

This completes the proof.

Since by the transpose property, the absolute value of is also equal to the volume of the paralellepiped determined by the columns of as well.

Example(Length)

A matrix is just a number In this case, the parallelepiped determined by its one row is just the interval (or if ). The “volume” of a region in is just its length, so it is clear in this case that

Example(Area)

When is a matrix, its rows determine a parallelogram in The “volume” of a region in is its area, so we obtain a formula for the area of a parallelogram: it is the determinant of the matrix whose rows are the vectors forming two adjacent sides of the parallelogram.

It is perhaps surprising that it is possible to compute the area of a parallelogram without trigonometry. It is a fun geometry problem to prove this formula by hand. [Hint: first think about the case when the first row of lies on the -axis.]

You might be wondering: if the absolute value of the determinant is a volume, what is the geometric meaning of the determinant without the absolute value? The next remark explains that we can think of the determinant as a signed volume. If you have taken an integral calculus course, you probably computed negative areas under curves; the idea here is similar.

The theorem on determinants and volumes tells us that the absolute value of the determinant is the volume of a paralellepiped. This raises the question of whether the sign of the determinant has any geometric meaning.

A matrix is just a number In this case, the parallelepiped determined by its one row is just the interval if and it is if In this case, the sign of the determinant determines whether the interval is to the left or the right of the origin.

For a matrix with rows the sign of the determinant determines whether is counterclockwise or clockwise from That is, if the counterclockwise angle from to is less than then the determinant is positive; otherwise it is negative (or zero).

For example, if then the counterclockwise rotation of by is by this example in Section 4.3, and

On the other hand, the clockwise rotation of by is and

For a matrix with rows the right-hand rule determines the sign of the determinant. If you point the index finger of your right hand in the direction of and your middle finger in the direction of then the determinant is positive if your thumb points roughly in the direction of and it is negative otherwise.

In higher dimensions, the notion of signed volume is still important, but it is usually defined in terms of the sign of a determinant.

Subsection5.3.3Volumes of Regions

Let be an matrix with columns and let be the associated matrix transformation Then and so takes the unit cube to the parallelepiped determined by

Since the unit cube has volume and its image has volume the transformation scaled the volume of the cube by a factor of To rephrase:

If is an matrix with corresponding matrix transformation and if is the unit cube in then the volume of is

The notation means the image of the region under the transformation In set builder notation, this is the subset

In fact, scales the volume of any region in by the same factor, even for curvy regions.

Theorem

Let be an matrix, and let be the associated matrix transformation If is any region in then

Proof

Let be the unit cube, let be the columns of and let be the paralellepiped determined by these vectors, so and For we let be the cube with side lengths i.e., the paralellepiped determined by the vectors and we define similarly. By the second defining property, takes to The volume of is (we scaled each of the standard vectors by a factor of ) and the volume of is (for the same reason), so we have shown that scales the volume of by

By the first defining property, the image of a translate of is a translate of

Since a translation does not change volumes, this proves that scales the volume of a translate of by

At this point, we need to use techniques from multivariable calculus, so we only give an idea of the rest of the proof. Any region can be approximated by a collection of very small cubes of the form The image is then approximated by the image of this collection of cubes, which is a collection of very small paralellepipeds of the form

The volume of is closely approximated by the sum of the volumes of the cubes; in fact, as goes to zero, the limit of this sum is precisely Likewise, the volume of is equal to the sum of the volumes of the paralellepipeds, take in the limit as The key point is that the volume of each cube is scaled by Therefore, the sum of the volumes of the paralellepipeds is times the sum of the volumes of the cubes. This proves that

The above theorem also gives a geometric reason for multiplicativity of the (absolute value of the) determinant. Indeed, let and be matrices, and let be the corresponding matrix transformations. If is the unit cube, then

On the other hand, the matrix for the composition is the product so