Linear algebra, determinants, and transposes

In summary, the determinant of a matrix is the sum of the products of its diagonals minus the products of its antidiagonals. When transposing a matrix, these products change.
  • #1
hgj
15
0
Okay, I need to prove that det(A^t) = det(A). I can see that it's true because I know columns and rows are interchangable (meaning you can use columns or rows when taking determinants), but I don't know how to prove this fact. Any help would be very appreciated.
 
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  • #2
hgj said:
Okay, I need to prove that det(A^t) = det(A). I can see that it's true because I know columns and rows are interchangable (meaning you can use columns or rows when taking determinants), but I don't know how to prove this fact. Any help would be very appreciated.
Well, a determinant of a matrix is the sum of the products of its diagonals minus the products of its antidiagonals. How do these products change under a transpose?
 
  • #3
[tex]\det{A}=\sum_{i=1}^{m}\left(-1\right)^{i+j}a_{ij}\det{A_{ij}}[/tex]

Do you see what happens when you try to prove det(AT)=det(A) for 2x2 or 3x3 matrices? Use the definition.

Edit: To the above poster: Doesn't that definition only work for 3x3 matrices?
 
Last edited:
  • #4
JoAuSc: that's only for 3x3 matrices.

apmcavoy: I don't believe that formula helps, at least not in a straightforward manner.

hgj: what are you using as the definition of a determinant? And have you yet proven that the determinant is multiplicative?
 
  • #5
If A is an nxn matrix, then
detA = a11det(A11) - a21*det(A21) + ... + (-1)^(n+1)*an1*det(An1)
(sorry, I don't know how to make things subscripts on this, so the 11, 21,...,n1 are supposed to be subscripts)

That's the definition we're using for a determinant.
 
  • #6
hgj said:
If A is an nxn matrix, then
detA = a11det(A11) - a21*det(A21) + ... + (-1)^(n+1)*an1*det(An1)
(sorry, I don't know how to make things subscripts on this, so the 11, 21,...,n1 are supposed to be subscripts)

That's the definition we're using for a determinant.

Ok, that's the same thing I posted above. I suppose you could write it out like you did for both A and AT, and then rearrange and show they are equal.
 

1. What is linear algebra?

Linear algebra is a branch of mathematics that deals with the study of vector spaces and linear transformations between them. It involves the use of algebraic techniques to solve systems of linear equations and represent geometric concepts.

2. What are determinants?

Determinants are a mathematical concept used to measure the size or scaling factor of a linear transformation. In linear algebra, determinants are represented as a single number associated with a square matrix. They are useful in solving systems of linear equations, calculating inverse matrices, and determining the invertibility of a matrix.

3. How do you calculate determinants?

The most common method for calculating determinants is by using the cofactor expansion method, also known as the Laplace expansion. This involves breaking down a matrix into smaller matrices and solving for the determinant of each one. Another method is using Gaussian elimination to reduce the matrix to an upper triangular form, making it easier to calculate the determinant.

4. What is a transpose?

A transpose is an operation that involves flipping a matrix over its diagonal. This results in the rows becoming columns and vice versa. It is represented by adding a superscript "T" to the matrix. Transposes are useful in solving systems of linear equations, calculating inner products, and performing rotations and reflections in linear transformations.

5. How is linear algebra used in real life?

Linear algebra has many applications in various fields such as physics, engineering, economics, and computer science. It is used to solve systems of equations in electrical circuits, calculate optimal solutions in economics, and analyze data in machine learning algorithms. It is also used in computer graphics to create 3D models and animations, and in cryptography for encrypting and decrypting messages.

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