What are determinants useful for?

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In summary, determinants are useful for determining if a matrix is invertible and for finding the scale factor in geometric transformations. They are also used in multiple integrals and in other areas of higher mathematics such as solving eigenvalue equations and proving theorems. The determinant function is continuous, meaning that small changes in the matrix's entries will result in small changes in the determinant. This allows for the matrix to remain invertible even with small changes. The axiomatic explanation of determinants can be found in the book "The Linear Algebra a Graduate Student Ought to Know."
  • #1
StephenPrivitera
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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?
 
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  • #2
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.
 
  • #3
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
 
  • #4
In case you haven't gotten the idea yet, it's called the determinant because it determines a lot of things about the matrix.
 
  • #5


Hurkyl said:
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?
 
  • #6


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.

Code:
>>> 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.
 
  • #7


Take the matrix
[tex]A = \begin{pmatrix}x & 2 \\ 3 & 4\end{pmatrix}[/tex].

The determinant of A is [tex] \det A = 4x - 3*2 = 4x - 6[/tex]. This is a continuous function of x. You can solve [tex]\det A = 0[/tex] 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.
 
  • #8


Lord Crc said:
This is a continuous function of x.

When you say this, do you mean the epsilon delta definition of continuous?
 
  • #10


StephenPrivitera said:
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.
 

What are determinants useful for?

1. How are determinants used in linear algebra?

Determinants are used to solve systems of linear equations, find the inverse of a matrix, and determine the area and volume of geometric shapes.

2. Can determinants be used in other areas of science?

Yes, determinants are used in physics, engineering, computer science, and many other fields to solve problems involving multiple variables and equations.

3. How do determinants help in understanding the properties of matrices?

Determinants provide information about the properties of matrices such as their rank, invertibility, and whether they are singular or nonsingular.

4. Are determinants only useful for matrices?

No, determinants can also be used for other mathematical objects such as tensors and linear transformations.

5. Can determinants be used to solve real-world problems?

Yes, determinants are commonly used to solve real-world problems in fields such as economics, biology, and statistics.

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