What's an intuitive way to think about the determinant?

In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t have an inverse. I can find the determinant of a $2\times 2$ matrix by the formula. Our teacher showed us how to compute the determinant of an $n \times n$ matrix by breaking it up into the determinants of smaller matrices. Apparently there is a way by summing over a bunch of permutations. But the notation is really hard for me and I don’t really know what’s going on with them anymore. Can someone help me figure out what a determinant is, intuitively, and how all those definitions of it are related?

$\begingroup$ I just wanted this question to be in the archives, because it's a perennial one that admits a better response than is in most textbooks. $\endgroup$

$\begingroup$ Hehe, you're going against your own suggestion of asking questions that you actually want answered!! I'm teasing though, I understand your motivation. Can we set a precedent of making seeded questions CW? I kinda like that idea, I will propose it on Meta. I am rambling. $\endgroup$

$\begingroup$ In case somebody was curious about the trace form of this question, it is more difficult, and is the subject of one of my MO questions :D mathoverflow.net/questions/13526/… $\endgroup$

$\begingroup$ Those reading this forum in '16 may be interested in this video The Determinant which is part of a set of videos that give some very nice insight into intuitive understanding of linear algebra (the essence of it rather) $\endgroup$

18 Answers 18

Your trouble with determinants is pretty common. They’re a hard thing to teach well, too, for two main reasons that I can see: the formulas you learn for computing them are messy and complicated, and there’s no “natural” way to interpret the value of the determinant, the way it’s easy to interpret the derivatives you do in calculus at first as the slope of the tangent line. It’s hard to believe things like the invertibility condition you’ve stated when it’s not even clear what the numbers mean and where they come from.

Rather than show that the many usual definitions are all the same by comparing them to each other, I’m going to state some general properties of the determinant that I claim are enough to specify uniquely what number you should get when you put in a given matrix. Then it’s not too bad to check that all of the definitions for determinant that you’ve seen satisfy those properties I’ll state.

The first thing to think about if you want an “abstract” definition of the determinant to unify all those others is that it’s not an array of numbers with bars on the side. What we’re really looking for is a function that takes N vectors (the N columns of the matrix) and returns a number. Let’s assume we’re working with real numbers for now.

Remember how those operations you mentioned change the value of the determinant?

1. Switching two rows or columns changes the sign.
2. Multiplying one row by a constant multiplies the whole determinant by that constant.
3. The general fact that number two draws from: the determinant is linear in each row. That is, if you think of it as a function $\det: \mathbb^ \rightarrow \mathbb$ , then $$ \det(a \vec v_1 +b \vec w_1 , \vec v_2 ,\ldots,\vec v_n ) = a \det(\vec v_1,\vec v_2,\ldots,\vec v_n) + b \det(\vec w_1, \vec v_2, \ldots,\vec v_n),$$ and the corresponding condition in each other slot.
4. The determinant of the identity matrix $I$ is $1$ .

I claim that these facts are enough to define a unique function that takes in N vectors (each of length N) and returns a real number, the determinant of the matrix given by those vectors. I won’t prove that, but I’ll show you how it helps with some other interpretations of the determinant.

In particular, there’s a nice geometric way to think of a determinant. Consider the unit cube in N dimensional space: the set of N vectors of length 1 with coordinates 0 or 1 in each spot. The determinant of the linear transformation (matrix) T is the signed volume of the region gotten by applying T to the unit cube. (Don’t worry too much if you don’t know what the “signed” part means, for now).

How does that follow from our abstract definition?

Well, if you apply the identity to the unit cube, you get back the unit cube. And the volume of the unit cube is 1.

If you stretch the cube by a constant factor in one direction only, the new volume is that constant. And if you stack two blocks together aligned on the same direction, their combined volume is the sum of their volumes: this all shows that the signed volume we have is linear in each coordinate when considered as a function of the input vectors.

Finally, when you switch two of the vectors that define the unit cube, you flip the orientation. (Again, this is something to come back to later if you don’t know what that means).

So there are ways to think about the determinant that aren’t symbol-pushing. If you’ve studied multivariable calculus, you could think about, with this geometric definition of determinant, why determinants (the Jacobian) pop up when we change coordinates doing integration. Hint: a derivative is a linear approximation of the associated function, and consider a “differential volume element” in your starting coordinate system.

It’s not too much work to check that the area of the parallelogram formed by vectors $(a,b)$ and $(c,d)$ is $\Big|<>^_\Big|$ either: you might try that to get a sense for things.

$\begingroup$ Great answer. We were taught the determinant as the generalized volume function in our algebra class. $\endgroup$

$\begingroup$ Just out of curiosity, who are you talking to with the first sentence? Didn't you ask the question. Either way, I point students to this Q (&A) all the time for determinant help. $\endgroup$

$\begingroup$ @TheChaz this question was asked near the beginnings of Math.SE, when there was a need to populate the site with questions before it was opened up to 'the public'. In any case, answering your own questions is explicitly encouraged nowadays. $\endgroup$

$\begingroup$ To see that the geometric interpretation (volume of the image of cube) satisfies the multilinearity property it is not enough to stack two "aligned" blocks, this is again the multiplication of a column by a scalar. You should deal with two "not aligned" blocks produced by changing a single column vector, and see that the sum of their volumes is the volume of the block obtained putting the sum of the two vectors. This is not so easy to catch. $\endgroup$

$\begingroup$ To be blunt, if the question was coming from a real questioner, this answer would not help. You've introduced more likely alien notation ($\mathbb^n^2$), given a geometric interpretation that is similar to the one in most texts (area of a parallelogram) and that doesn't give any intuition at all. After reading this answer I have no better idea how someone would have plausibly come to the notion of a determinant in the first place. A bunch of people that already understand the concept appreciating the answer is a poor test for deciding the educational value of an explanation. $\endgroup$

You could think of a determinant as a volume. Think of the columns of the matrix as vectors at the origin forming the edges of a skewed box. The determinant gives the volume of that box. For example, in 2 dimensions, the columns of the matrix are the edges of a rhombus.

You can derive the algebraic properties from this geometrical interpretation. For example, if two of the columns are linearly dependent, your box is missing a dimension and so it's been flattened to have zero volume.

$\begingroup$ If I may, I would add to this answer (which I think is a very good one) in two minor aspects. First, a determinant also has a sign, so we want the concept of oriented volume. (This is somewhat tricky, but definitely important, so you might as well have it in mind when you're learning about "right hand rules" and such.) Second, I think better than a volume is thinking of the determinant as the multiplicative change in volume of a parallelopiped under the linear transformation. (Of course you can always take the first one to be the unit n-cube and say that you are just dividing by one.) $\endgroup$

$\begingroup$ I'm ten years late, but here is a video by 3blue1brown on the determinant which uses the same geometric interpretation. $\endgroup$

In addition to the answers, above, the determinant is a function from the set of square matrices into the real numbers that preserves the operation of multiplication: \begin\det(AB) = \det(A)\det(B) \end and so it carries $some$ information about square matrices into the much more familiar set of real numbers.

The determinant function maps the identity matrix $I$ to the identity element of the real numbers ($\det(I) = 1$.)

Which real number does not have a multiplicative inverse? The number 0. So which square matrices do not have multiplicative inverses? Those which are mapped to 0 by the determinant function.

What is the determinant of the inverse of a matrix? The inverse of the determinant, of course. (Etc.)

This "operation preserving" property of the determinant explains some of the value of the determinant function and provides a certain level of "intuition" for me in working with matrices.

Here is a recording of my lecture on the geometric definition of determinants:

It has elements from the answers by Jamie Banks and John Cook, and goes into details in a leisurely manner.

I too find the way determinants are treated in exterior algebra most intuitive. The definition is given on page 46 of Landsberg's "Tensors: Geometry and Applications". Two examples below will tell you everything you need to know.

Say, you are give a matrix $$A=\begina&b\\c&d\end$$ and asked to compute its determinant. You can think of the matrix as a linear operator $f:\mathbb R^2\to\mathbb R^2$ defined by

If you define the standard basis vector by $e_1=\begin1\\0\end$ and $e_2=\begin0\\1\end$, you can then define $f$ by the values it assumes on the basis vectors: $f(e_1)=ae_1+ce_2$ and $f(e_2)=be_1+de_2$.

The linear operator $f$ is extended to bivectors by $$f(e_1\wedge e_2)=f(e_1)\wedge f(e_2).$$

Then you can write

$$f(e_1\wedge e_2)=(ae_1+ce_2)\wedge(be_1+de_2)=(ad-bc)e_1\wedge e_2,$$

where I used distributivity and anticommutativity of the wedge product (the latter implies $a\wedge a=0$ for any vector $a$). So, we get the determinant as a scalar factor in the above equation, that is

$$f(e_1\wedge e_2)=\det(A)\,e_1\wedge e_2.$$

The same procedure works for 3-by-3 matrices, you just need to use a trivector. Say, you are given $$B=\begina_&a_&a_\\a_&a_&a_\\a_&a_&a_\end.$$

It defines a linear operator $g:\mathbb R^3\to \mathbb R^3$

for which we have

$$g(e_1)=a_e_1+a_e_2+a_e_3,\quad g(e_2)=a_e_1+a_e_2+a_e_3,\quad g(e_3)=a_e_1+a_e_2+a_e_3$$

on the standard basis $e_1=\begin1\\0\\0\end$, $e_2=\begin0\\1\\0\end$, $e_3=\begin0\\0\\1\end$. The operator $g$ is extended to trivectors by $$g(e_1\wedge e_2\wedge e_3)=g(e_1)\wedge g(e_2)\wedge g(e_3),$$

$$g(e_1\wedge e_2\wedge e_3)=(a_e_1+a_e_2+a_e_3)\wedge(a_e_1+a_e_2+a_e_3)\wedge(a_e_1+a_e_2+a_e_3).$$

If you then follows the rules of $\wedge$ such as distributivity, anticommutativity, and associativity, you get $$g(e_1\wedge e_2\wedge e_3)=\det(B)\, e_1\wedge e_2\wedge e_3.$$

It works in exactly the same way in higher dimensions.

For the record I'll try to give a reply to this old question, since I think some elements can be added to what has been already said.

Even though they are basically just (complicated) expressions, determinants can be mysterious when first encountered. Questions that arise naturally are: (1) how are they defined in general?, (2) what are their important properties?, (3) why do they exist?, (4) why should we care?, and (5) why does their expression get so huge for large matrices?

Since $2\times2$ and $3\times3$ determinants are easily defined explicitly, question (1) can wait. While (2) has many answers, the most important ones are, to me: determinants detect (by becoming 0) the linear dependence of $n$ vectors in dimension $n$, and they are an expression in the coordinates of those vectors (rather than for instance an algorithm). If you have a family of vectors that depend (or at least one of them depends) on a parameter, and you need to know for which parameter values they are linearly dependent, than trying to use for instance Gaussian elimination to detect linear dependence can run into trouble: one might need assumptions on the parameter to assure some coefficient is nonzero, and even then dividing by it gives very messy expressions. Provided the number of vectors equals the dimension $n$ of the space, taking a determinant will however immediately transform the question into an equation for the parameter (which one may or may not be capable of solving, but that is another matter). This is exactly how one obtains an equation in eigenvalue problems, in case you've seen those. This provides a first answer to (4). (But there is a lot more you can do with determinants once you get used to them.)

As for question (3), the mystery of why determinants exist in the first place can be reduced by considering the situation where one has $n-1$ given linearly independent vectors, and asks when a final unknown vector $\vec x$ will remain independent from them, in terms of its coordinates. The answer is that it usually will, in fact always unless $\vec x$ happens to be in the linear span $S$ of those $n-1$ vectors, which is a subspace of dimension $n-1$. For instance, if $n=2$ (with one vector $\vec v$ given) the answer is "unless $\vec x$ is a scalar multiple of $\vec v$". Now if one imagines a fixed (nonzero) linear combination of the coordinates of $\vec x$ (the technical term is a linear form on the space), then it will become $0$ precisely when $\vec x$ is in some subspace of dimension $n-1$. With some luck, this can be arranged to be precisely the linear span $S$. (In fact no luck is involved: if one extends the $n-1$ vectors by one more vector to a basis, then expressing $\vec x$ in that basis and taking its final coordinate will define such a linear form; however you can ignore this argument unless you are particularly suspicious.) Now the crucial observation is that not only does such a linear combination exist, its coefficients can be taken to be expressions in the coordinates of our $n-1$ vectors. For instance in the case $n=2$ if one puts $\vec v=$ and $\vec x=$, then the linear combination $-bx_1+ax_2$ does the job (it becomes 0 precisely when $\vec x$ is a scalar multiple of $\vec v$), and $-b$ and $a$ are clearly expressions in the coordinates of $\vec v$. In fact they are linear expressions. For $n=3$ with two given vectors, the expressions for the coefficients of the linear combination are more complicated, but they can still be explicitly written down (each coefficient is the difference of two products of coordinates, one form each vector). These expressions are linear in each of the vectors, if the other one is fixed.

Thus one arrives at the notion of a multilinear expression (or form). The determinant is in fact a multilinear form: an expression that depends on $n$ vectors, and is linear in each of them taken individually (fixing the other vectors to arbitrary values). This means it is a sum of terms, each of which is the product of a coefficient, and of one coordinate each of all the $n$ vectors. But even ignoring the coefficients, there are many such terms possible: a whopping $n^n$ of them!

However, we want an expression that becomes $0$ when the vectors are linearly dependent. Now the magic (sort of) is that even the seemingly much weaker requirement that the expression becomes $0$ when two successive vectors among the $n$ are equal will assure this, and it will moreover almost force the form of our expression upon us. Multilinear forms that satisfy this requirement are called alternating. I'll skip the (easy) arguments, but an alternating form cannot involve terms that take the same coordinate of any two different vectors, and they must change sign whenever one interchanges the role of two vectors (in particular they cannot be symmetric with respect to the vectors, even though the notion of linear dependence is symmetric; note that already $-bx_1+ax_2$ is not symmetric with respect to interchange of $(a,b)$ and $(x_1,x_2)$). Thus any one term must involve each of the $n$ coordinates once, but not necessarily in order: it applies a permutation of the coordinates $1,2,\ldots,n$ to the successive vectors. Moreover, if a term involves one such permutation, then any term obtained by interchanging two positions in the permutation must also occur, with an opposite coefficient. But any two permutations can be transformed into one another by repeatedly interchanging two positions; so if there are any terms at all, then there must be terms for all $n!$ permutations, and their coefficients are all equal or opposite. This explains question (5), why the determinant is such a huge expression when $n$ is large.

Finally the fact that determinants exist turns out to be directly related to the fact that signs can be associated to all permutations in such a way that interchanging entries always changes the sign, which is part of the answer to question (3). As for question (1), we can now say that the determinant is uniquely determined by being an $n$-linear alternating expression in the entries of $n$ column vectors, which contains a term consisting of the product of their coordinates $1,2,\ldots,n$ in that order (the diagonal term) with coefficient $+1$. The explicit expression is a sum over all $n!$ permutations, the corresponding term being obtained by applying those coordinates in permuted order, and with the sign of the permutation as coefficient. A lot more can be said about question (2), but I'll stop here.