How to prove that the determinant of K is also zero without using eigenvalues?

In summary, to prove that the determinant of matrix K is 0 when b * K = 0, you can use the fact that the solution to KTb = 0 is not unique, which means that the determinant of KT must be 0. This can also be proved using Kramer's rule by substituting a row of K with 0 and showing that the determinant of the resulting matrix is 0.
  • #1
EmmaSaunders1
45
0
Hi,

if vector b * matrix K = 0 (bK=o) what methods can one use to show that the determinant of K is therefore also zero, without using eigenvalues.

I have a feeling I am over complicating this.

Knd regards

Emma
 
Physics news on Phys.org
  • #2
I assume b != 0. What else are you starting with? Are you allowed to use the fact that, for any square matrix K with determinant != 0, the equation Kx = y has a unique solution? If so, it is obvious that the solution to KTb = 0 is not unique, since KT0 = 0 also. Thus, the determinant of KT is 0, and since transposition doesn't change the determinant, the determinant of K must also be 0.
 
  • #3
Sorry that's correct that b!=0

using for any square matrix K with determinant != 0, the equation Kx = y has a unique solution - is fine.

Im almost understand where your heading with this - would you please however clarify two steps;

you say KTb = 0 is not unique, since KT0 = 0 also. Thus, the determinant of KT is 0.

How do you know the value of the determinant from that statement alone,

Also is bK, the same as KTb, I noticed you switched the order of multiplication there.

Thanks for your help
 
  • #4
Sorry, I should have written KTbT. I'm using the turnover rule for matrix multiplication: for any two matrices A and B, (AB)T = BTAT. In this case b is a row vector, a 1 x n matrix, and bT is a column vector, an n x 1 matrix. The only reason for doing it that way is that the uniqueness theorem is usually stated in terms of left multiplication and column vectors, but of course it also holds for right multiplication and row vectors.

you say KTb = 0 is not unique, since KT0 = 0 also. Thus, the determinant of KT is 0.

How do you know the value of the determinant from that statement alone
If the determinant of K (or KT) is not 0, the solution would be unique. Since the solution is not unique, the determinant must be 0.
 
  • #5
Thats great - I really appreciate your help!
 
  • #6
It can be proved directly using Kramer's rule. Write K=[k1;...;kn] (the kj are the rows) and let Kj be the matrix K with row j replaced with 0. Obviously det(Kj)=0 by row expansion.

Also 0=b*K=b1*k1+...+bn*kn, so det(Kj)=det([k1;...;b1*k1+...+bn*kn;...;kn])=det([k1;...;bj*kj;...;kn])=bj*det(K). Choose any j such that bj<>0 and this shows that det(K)=0.
 

1. What is a determinant in simple algebra?

A determinant in simple algebra is a numerical value that can be calculated for a square matrix. It is used to determine whether a matrix has a unique solution, and if so, what that solution is.

2. How is a determinant calculated?

A determinant can be calculated by using a specific formula depending on the size of the matrix. For a 2x2 matrix, the determinant is calculated by subtracting the product of the top left and bottom right elements from the product of the top right and bottom left elements. For larger matrices, the Laplace expansion method or Gaussian elimination method can be used.

3. What does the determinant tell us about a matrix?

The determinant of a matrix can tell us several things. If the determinant is equal to 0, it means that the matrix does not have a unique solution, and it is singular. If the determinant is non-zero, it means the matrix has a unique solution and is invertible. The sign of the determinant can also indicate whether the solution will be positive or negative.

4. Why is the determinant important in linear algebra?

The determinant is important in linear algebra because it is used to determine the invertibility of a matrix and find solutions to systems of linear equations. It is also used in various other calculations, such as finding the area and volume of shapes and determining the eigenvalues of a matrix.

5. Are there any real-life applications of determinants in simple algebra?

Yes, there are several real-life applications of determinants in simple algebra. For example, they are used in computer graphics to calculate the orientation of 3D objects, in physics to calculate the moment of inertia of objects, and in economics to solve systems of linear equations in market equilibrium models.

Similar threads

Replies
3
Views
1K
  • Precalculus Mathematics Homework Help
Replies
1
Views
448
  • Quantum Physics
Replies
2
Views
907
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
877
  • Calculus and Beyond Homework Help
Replies
24
Views
616
Replies
34
Views
2K
  • Linear and Abstract Algebra
Replies
3
Views
928
  • Computing and Technology
Replies
2
Views
137
  • Precalculus Mathematics Homework Help
Replies
1
Views
636
  • Linear and Abstract Algebra
Replies
10
Views
2K
Back
Top