Why are determinants so important

In fact, I'm afraid if I tried to memorize it, I might forget something else important, like how to combine like terms in algebra. The above procedure generalizes to larger determinants. That's too messy to write down. But if you had to, you could do it. Usually, though, we'd offload such an ugly and boring calculation to a computer.

Here is one cute analysis application. Warning: I am basically just speculating, and not commenting with actual knowledge of the history.

I suspect that a lot of the nineteenth century work on determinants was motivated by invariant theory. These invariants often have a basis consisting of various sorts of determinantal expressions, and if you want to prove finite generation, you have to construct ways of taking certain determinantal expressions and writing them in terms of other determinantal expressions.

Within a decade or so of Hilbert's paper, people generally lost interest in constructive invariant theory. After all, the abstract methods answered the most interesting questions and seemed much more likely than constructive methods to answer the most interesting remaining questions.

I have wondered whether all the work on syzygies of determinantal varieties actually reduces to identities which were well known to the right people in the 19th century. The area of an arbitrary triangle is most easily computed from its vertex coordinates as a 3x3 determinant no square roots, trig, etc.

This generalizes to 3D volume of a tetrahedron and higher dimensions. The cross product of two vectors a,b essential tool in mechanics is easily understood and memorized once one observes that its coordinates are 2x2 minor determinants of a 2x3 array formed by a and b.

This too generalizes and provides a definition for cross product of n-1 vectors in n dimensions. More generally, determinants combined with homogeneous coordinates are all one needs to derive elegant formulas for the basic operations of n-dimensional projective geometry, such as plane through 3 points, intersection of 3 planes. They also provide a coordinate representation Pluecker coordinates for lines in 3-space, or more generally for k-dim subspaces of n-dim prijective space.

The sign of the determinant of a matrix tells whether the rows are a left- or right-handed frame, and whether the corresponding linear map preserves or reverses orientations. Determinants are more efficient than Gaussian elimination to compute the inverse of a 2x2 or 3x3 matrix, perhaps even 4x4. Unlike standard G. For the reasons of reputation points, I am unable to comment on KConrad's post. I up-voted his response, however, because he does make a great point about the Muir book s.

From the realm of probability there are determinental and permanental processes. Terry Tao has a nice post about determinental processes here. For instance. I've not worked with these processes myself but I've heard enough seminar talks using them to say there is plenty of interest out there.

Just an example from applied statistics where determinants are unavoidable because they are used to define the relevant concepts. One way of justifying this is that the requirement is equivalent to minimizing the volume of confidence ellipsoids, calculated under the normal model.

Mostly, numerical optimization is used to construct such designs. Of course, they will have pretty ugly condition numbers. Sign up to join this community. The best answers are voted up and rise to the top. Why were matrix determinants once such a big deal? Ask Question. Asked 11 years, 2 months ago. Active 3 years, 6 months ago. Viewed 35k times. So here's my question: a What are examples of cool tricks you can use matrix determinants for? I think it would be useful to generalize the original series of questions with: c What significance do matrix determinants have for other branches of mathematics?

Improve this question. The classical approach to invariant theory goes through Cappelli's formula and other determinant relations. Determinant ideals play a big role in the theory of modules over commutative rings. Resultants and discriminants the oldest and main method of solving systems of nonlinear polynomial equations with arbitrary precision - at least in theory are defined as particular determinants.

Axler's Down with Determinants paper which should have been called Down with Characteristic Polynomials implicitly takes the k[x]-module approach. To me, this doctrinal approach appears just as fruitless as the attempts to base real analysis on "constructible" numbers only since countably many reals expressible in a finite way "should be sufficient".

One should elucidate its algebraic, geometric and combinatorial structure rather than try to banish its use. As for computing eigenvalues, the preferred tools in applications are Krylov subspace methods. Though you don't usually see it mentioned in books on numerical analysis, their structure is very much based on notions that fall out of this k[x]-module approach.

Show 8 more comments. Active Oldest Votes. Change of variable in an integral. Isn't the Jacobian of a transformation a determinant? The Wronskian of solutions of a linear ODE is a determinant. That is. Determinant of Cofactor Matrix.

Property of Invariance. Examples Problems on Properties of Determinants. Some important example on properties of determinants are given below:. Using Properties of Determinant, Prove That. Solution : With the help of the invariance and scalar multiple properties of determinant we can prove the above- given determinant.

Solution: Interchanging the rows and columns across the diagonals by making use of reflection property and then using the switching property of determination we can get the desired outcome. Interchanging rows and columns across the diagonals. Quiz Time. Multiplied by row. Multiplied to column.

Divided to row. Are there any real life applications of determinants? Is there a really good motivating example or explanation which will hook students into this topic? In linear algebra, where should determinants be placed?

Like I said in my comment - in some literature it is at the beginning whilst in others it is bolted on at the end. I like the idea of checking if vectors are independent by using determinants so think they should be placed before independence of vectors.

What do you think? If you teach a linear algebra course where do you place this topic. You are witnessing a shift in emphasis away from determinants.

This does not mean they are unimportant; on the contrary, they are quite important — think of the change of variable formula in multiple integrals, for instance — but by introducing them too early in linear algebra courses, and spending too much time on their properties, we have encouraged students to use determinants where their use is not appropriate, such as in solving linear systems.

In fact, the determinant function is the unique multilinear alternating function when evaluated on the Identity matrix, treating columns as vectors, yields unity.

From this characterization, one can formally derive the admittedly horrendous-looking combinatorial formula. This is preferable in my opinion than stating the formula and showing that said formula satisfies properties A-Z.

The determinant shows up in a surprising number of places and for this reason alone is important. I think though it might be an excellent place to introduce the idea that mathematical objects are not necessarily important for what they are but, rather, the properties they satisfy e. If you think linear algebra is a good place to introduce the basic concepts of abstract mathematics I think the determinant does this quite nicely.

The description of the determinant above is elementary enough that anyone taking linear algebra should be able to understand it and proving subsequent theorems about determinants goes significantly smoother with this characterization as opposed to trying to pound it out with the combinatorial formula.