Linear Algebra Problem: Proving det(A) = det(B) for Related Matrices A and B

In summary, in order to prove that det(A) = det(B), we need to use the fact that the determinant of a product of two matrices is equal to the product of their determinants, and the fact that the determinant of an invertible matrix is equal to the reciprocal of its determinant. With these two facts, we can show that det(A) = det(B) for any two n x n matrices related by the equation (P^-1)(A)(P) = B, where P is another n x n matrix.
  • #1
DWill
70
0

Homework Statement


Let A and B be two n x n matrices that are related by the equation (P^-1)(A)(P) = B, where P is another n x n matrix. Prove that det(A) = det(B).


Homework Equations





The Attempt at a Solution


I'm thinking the first step might be to come up with general forms of A and B that are related by the above equation? I've been trying to do that and not been successful so far. Any ideas? thanks
 
Physics news on Phys.org
  • #2
Why not just 'compute' det(B)?
 
  • #3
DWill said:

Homework Statement


Let A and B be two n x n matrices that are related by the equation (P^-1)(A)(P) = B, where P is another n x n matrix. Prove that det(A) = det(B).


Homework Equations





The Attempt at a Solution


I'm thinking the first step might be to come up with general forms of A and B that are related by the above equation? I've been trying to do that and not been successful so far. Any ideas? thanks
Two basic facts you should know (and use them in this exercise):
The equality
det(A.B)=det(A).det(B)
is true for any two nxn matrices.
And we have for any invertible matrix
det(A^-1)= ?
(I think you should be able to guess the result using the definition of the inverse and the above equation.)
That's all you need to know in order to solve this one.
 

What is linear algebra?

Linear algebra is a branch of mathematics that deals with linear equations and their representations in vector spaces. It involves the study of linear transformations, matrices, and systems of linear equations.

What are the applications of linear algebra?

Linear algebra has numerous applications in various fields such as physics, engineering, computer science, economics, and statistics. It is used for solving systems of equations, analyzing data, and creating computer graphics, among others.

What are the basic concepts of linear algebra?

The basic concepts of linear algebra include vectors, matrices, vector spaces, linear transformations, eigenvalues and eigenvectors, and determinants. These concepts are used to solve linear equations and represent real-world problems mathematically.

What is a linear algebra problem?

A linear algebra problem is a mathematical problem that involves solving linear equations or systems of linear equations using various methods and techniques from linear algebra. These problems often require manipulating matrices, vectors, and other concepts from linear algebra to find a solution.

What are some common techniques for solving linear algebra problems?

Some common techniques for solving linear algebra problems include Gaussian elimination, matrix multiplication, inverse matrices, and vector operations. Other techniques, such as determinants and eigenvalues, are used for more advanced problems. The choice of technique depends on the specific problem at hand.

Similar threads

  • Calculus and Beyond Homework Help
Replies
25
Views
2K
  • Calculus and Beyond Homework Help
Replies
11
Views
1K
  • Calculus and Beyond Homework Help
Replies
17
Views
1K
  • Calculus and Beyond Homework Help
Replies
14
Views
594
  • Calculus and Beyond Homework Help
Replies
5
Views
4K
  • Calculus and Beyond Homework Help
Replies
5
Views
2K
  • Calculus and Beyond Homework Help
Replies
10
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
5K
  • Calculus and Beyond Homework Help
Replies
28
Views
4K
  • Calculus and Beyond Homework Help
Replies
1
Views
458
Back
Top