What are determinants useful for?

[StephenPrivitera]

I want to learn about determinants, but I'm just getting so bored studying the hundred and one ways to manipulate the determinant to find its value especially when they don't seem to have much value. What are determinants useful for?

 

[Hurkyl]

The first use most people will encounter to determinants is to tell whether or not a matrix is invertible (iow, if a system of equations has a unique solution).

There is a geometric meaning to determinants, though; recall that square matrices are linear transformations from Rⁿ to itself; however this linear transformation does not necessarily preserve shape... the determinant is the factor by which a matrix scales n-volumes.

For instance, if the determinant of a 2x2 matrix is 5, then if you use the 2x2 matrix to transform the plane, all areas will be multiplied by 5.

Or, if the determinant of a 3x3 matrix is -1/4, then if you use the 3x3 matrix to transform 3-space, all volumes will be multiplied by 1/4 and have their orientation reversed. (the negative works like a reflection)

If you've had multivariable calculus yet, the determinant is how the change of variables formula for multiple integrals works. We imagine "dx dy" to be the area of an infinitessimal parallelogram. When we change coordinates to (u, v), we get a new infinitessimal element "du dv". Coordinate changes are locally linear, so at each point there is a matrix that represents the local transformation (the Jacobian) and the determinant of the Jacobian is the scale factor of the transformation, thus it is the determinant of the Jacobian that appears in the new integrand.

Another use of the determinant that appears in higher mathematics is that the determinant is a continuous function of a matrix, and knowing the properties of the determinant allow some one line proofs of theorems that would be otherwise difficult.

 

[lethe]

determinants are used to solve the eigenvalue equation, determinants are used to find the volume form of a riemannian space, determinants are used to guarantee the invertibility of matrix groups, determinants are used to check whether the bijectivity of a diffeomorphism

 

[Tyger]

In case you haven't gotten the idea yet, it's called the determinant because it determines a lot of things about the matrix.

 

[fk378]

Another use of the determinant that appears in higher mathematics is that the determinant is a continuous function of a matrix, and knowing the properties of the determinant allow some one line proofs of theorems that would be otherwise difficult.

Can you explain how the determinant function is continuous? And also, my professor mentions how *because* the determinant is continuous, we can change the entries in small amounts and still have the matrix be invertible. Huh?

 

[mikeph]

Think of an M-by-N matrix containing real elements. Then the function to calculate the determinant is a continuous function of the M*N variables.

eg.

>>> det([1,2;3,4])
ans =
    -2
>>> det([1,2;3,4.01])
ans =
   -1.9900
>>> det([1,2;3,4.0001])
ans =
   -1.9999

Which can go on and on, since it is just formed by multiplying and adding elements.

In general for a 2x2,
f(a,b,c,d) = det|a, b; c, d| = a*d - b*c
and all the derivatives df/da, df/db, etc. exist.

 

[Lord Crc]

Take the matrix
$$A = \begin{pmatrix}x & 2 \\ 3 & 4\end{pmatrix}$$.

The determinant of A is $$\det A = 4x - 3*2 = 4x - 6$$. This is a continuous function of x. You can solve $$\det A = 0$$ with respect to x to find for which value of x the matrix A is not invertible. By extension, for any other value of x the matrix A is invertible.

 

[fk378]

This is a continuous function of x.

When you say this, do you mean the epsilon delta definition of continuous?

 

[Lord Crc]

Yes, see for instance http://mathworld.wolfram.com/Epsilon-DeltaProof.html for a proof of that.

 

[chiro]

I want to learn about determinants, but I'm just getting so bored studying the hundred and one ways to manipulate the determinant to find its value especially when they don't seem to have much value. What are determinants useful for?

There is a book I'm reading at the moment called "The Linear Algebra a Graduate Student Ought to know" 2nd edition which is published by Springer. You might be able to pick it up at your university library.

In the book it contains an axiomatic explanation of the determinant, how the "determinant" formula is essentially derived and its properties. Although I won't post the details I will say that the axioms you start off with deal with determinants of zero and identity matrices and properties of the determinant of specific row operations and what that does to the determinant. The book then uses these "axioms" to derive the inductive formula that is stated in all linear algebra textbooks.

If you are going to do graduate study or at least planning on it, or even if you want to understand what you are doing I recommend getting a hold of this book.