Linear Algebra Determinants Proof

In summary: Can you give me an example of when you would use this equation?As you said, a determinant is just a number. You treat them just like numbers. That's why I said let x=det(A). ##x^3=x## looks less mysterious.This equation is used when you want to find the determinant of a matrix.
  • #1
B18
118
0

Homework Statement


Prove that if A is an n x n matrix with the property A3=A, then det(A)=-1, det(A)=0, or det(A)=1

Homework Equations

The Attempt at a Solution


At first I started with the property A3=A
I then applied the determinant to both sides.
From this point I don't really see any properties that I can use to solve this equation for the det(A).
I understand how the solutions work. I am just unsure on how to achieve the proof of those solutions to the property.
Am I missing something simple here?
 
Last edited:
Physics news on Phys.org
  • #2
B18 said:

Homework Statement


Prove that if A is an n x n matrix with the property A3=A, then det(A)=-1, det(A)=0, or det(A)=1

Homework Equations

The Attempt at a Solution


At first I started with the property A3=A
I then applied the determinant to both sides.
From this point I don't really see any properties that I can use to solve this equation for the det(A).
Am I missing something simple here?

Did you apply the property that det(AB)=det(A)det(B)? A^3=AAA.
 
  • #3
Dick said:
Did you apply the property that det(AB)=det(A)det(B)? A^3=AAA.
Hi Dick, sorry. I did apply this property. I I'm just trying to get past this step.
 
  • #4
B18 said:
Hi Dick, sorry. I did apply this property. I I'm just trying to get past this step.

Then show me what you got after you applied the property.
 
  • #5
Dick said:
Then show me what you got after you applied the property.
det(A)*det(A)*det(A)=det(a)
(det(A))^3=det(A)
 
  • #6
B18 said:
det(A)*det(A)*det(A)=det(a)
3det(A)=det(A)

det(A)*det(A)*det(A) is not equal to 3det(A). It's equal to ##det(A)^3##. Let x=det(A). Then your equation is ##x^3=x##. Can you solve that equation for x?
 
  • #7
(det(A))3=det(A)
(det(A))3-det(A)=0
det(A)(det(A)2-1)=0
Thus:
det(A)=0
and det(A)2=1
det(A)=+/-1
 
  • #8
B18 said:
(det(A))3=det(A)
(det(A))3-det(A)=0
det(A)(det(A)2-1)=0
Thus:
det(A)=0
and det(A)2=1
det(A)=+/-1

Much better.
 
  • Like
Likes B18
  • #9
Is there a property that allows us to manipulate determinants this way? Or is it applicable because a determinate is a real number. Just need to build my proof format now. Really appreciate the help.
 
  • #10
B18 said:
Is there a property that allows us to manipulate determinants this way? Or is it applicable because a determinate is a real number. Just need to build my proof format now. Really appreciate the help.

As you said, a determinant is just a number. You treat them just like numbers. That's why I said let x=det(A). ##x^3=x## looks less mysterious.
 

1. What is a determinant in linear algebra?

A determinant is a mathematical operation that is defined for square matrices. It is a real number that can be computed from the entries of the matrix and provides important information about the properties of the matrix.

2. What is the importance of determinants in linear algebra?

Determinants are important in linear algebra because they provide valuable information about the properties of a matrix. For example, they can be used to determine the invertibility of a matrix, to solve systems of linear equations, and to calculate the area or volume of a geometric shape.

3. How do you prove properties of determinants in linear algebra?

To prove properties of determinants in linear algebra, one must use various techniques such as row operations, cofactor expansion, and the properties of determinants themselves. These techniques allow one to manipulate the entries of a matrix and show that a specific property holds true.

4. What are some common properties of determinants?

Some common properties of determinants include:

  • The determinant of the identity matrix is equal to 1
  • Multiplying a row or column of a matrix by a constant multiplies the determinant by the same constant
  • Swapping two rows or columns of a matrix changes the sign of the determinant
  • The determinant of a triangular matrix is equal to the product of its diagonal entries

5. Can determinants be negative?

Yes, determinants can be negative. The sign of a determinant depends on the number of row or column swaps that were performed during the computation. If an odd number of swaps were performed, the determinant will be negative, and if an even number of swaps were performed, the determinant will be positive.

Similar threads

Proving ##(cof ~A)^t ~A = (det A)I##
    • Calculus and Beyond Homework Help
Replies
4
Views
965
Proving statements about matrices | Linear Algebra
    • Calculus and Beyond Homework Help
Replies
25
Views
2K
Matrix concept Questions (invertibility, det, linear dependence, span)
    • Calculus and Beyond Homework Help
Replies
4
Views
949
Proving det(A) > 0 for A^3 = A + 1 over R using linear algebra
    • Calculus and Beyond Homework Help
Replies
17
Views
1K
Proving that determinants aren't linear transformations?
    • Calculus and Beyond Homework Help
Replies
5
Views
2K
Why is my calculation for the determinant of a matrix incorrect?
    • Calculus and Beyond Homework Help
Replies
6
Views
1K
Proof regarding determinant of block matrices
    • Calculus and Beyond Homework Help
Replies
5
Views
4K
Determinant Problem: Show $\det B = \det A$
    • Calculus and Beyond Homework Help
Replies
11
Views
1K
Eigenvalues of transpose linear transformation
    • Calculus and Beyond Homework Help
Replies
7
Views
2K
Do these two statements imply an underlying induction proof?
    • Calculus and Beyond Homework Help
Replies
7
Views
419

Hot Threads

Recent Insights

Back
Top