Prove the following euqlaity of determinants

  • Thread starter Jennifer1990
  • Start date
  • Tags
    Determinants
In summary, the determinant of a 3x3 matrix is the sum of the cofactors multiplied by the number in each entry of the original matrix.
  • #1
Jennifer1990
55
0

Homework Statement


Prove
det [a+p b+r c+s; d e f; g h i] = det [ a b c; d e f; g h i] + det [p r s; d e f; g h i]



Homework Equations


none


The Attempt at a Solution


i'm not sure how to prove this though its seems obviously true =S
 
Physics news on Phys.org
  • #2
Hi Jennifer1990! :smile:

General tip: if you can't see how to prove something, always go back to the definition. :wink:

Sooo … what is the definition of the determinant? :smile:
 
  • #3


A determinant is a function depending on n that associates a scalar, det(A), to an n×n square matrix A.

I don't see how this helps me tho.. I'm sorry T_T
 
  • #4


Jennifer1990 said:
A determinant is a function depending on n that associates a scalar, det(A), to an n×n square matrix A.

I don't see how this helps me tho.. I'm sorry T_T
That's a very general description of what a determinant is, but not a definition. Do you know how to calculate the determinant of a 3 x 3 matrix?
 
  • #5


To calculate the determinant of a 3x3 matrix, u take the sum of the cofactors multiplied by the number in each entry of the original matrix . which is then also multiplied by (-1)^n, where n represents the position of the entry
 
  • #6


Not quite. As you have written it, it appears that you are saying you have a sum of 9 terms for a 3 x 3 matrix, which is not true.

You can evaluate a 3 x 3 matrix by cofactors going across any row or down any column. So for your first matrix, one way of evaluating the determinant is to expand across the top row, resulting in (a + p)A11 - (b + r)A12 + (c + s)A13. The Aijs are the cofactors, meaning that each one is the determinant of the 2 x 2 matrix made up of the entries not in row i and column j. I have already taken into account the appropriate sign of each term.

Now, have at it...
 
  • #7


um actually, that's wat i meant o.o
 
  • #8


But that's not what you said, so it seemed to me that you didn't really understand how to calculate the determinant.
 
  • #9


ohh sorry.
so, um, are u asking me to apply that formula to the determinants in the question and show that the determinants are the same?
 
  • #11
(just got up :zzz:)

me too! :biggrin:
 
  • #12


Is there another way to prove this question other than finding the determinant for the matrices?
 
  • #13
Jennifer1990 said:
Is there another way to prove this question other than finding the determinant for the matrices?

but the question is about the values of the determinants!

so, noooo :rolleyes:

get on with it! :smile:
 

What does it mean to "prove the following equality of determinants"?

Proving the equality of determinants means to show that two determinants are equal to each other, using algebraic or geometric methods.

Why is it important to prove the equality of determinants?

Proving the equality of determinants is important because it helps us understand and verify the properties and relationships between different matrices.

What are the common methods used to prove the equality of determinants?

The two most common methods used to prove the equality of determinants are the expansion method and the row or column operations method.

Can the equality of determinants be proven for any size matrix?

Yes, the equality of determinants can be proven for any size matrix as long as both matrices have the same number of rows and columns.

What are some real-life applications of proving the equality of determinants?

Proving the equality of determinants is commonly used in fields such as physics, engineering, and computer graphics to solve problems involving systems of linear equations and geometric transformations.

Similar threads

  • Calculus and Beyond Homework Help
Replies
6
Views
517
  • Calculus and Beyond Homework Help
Replies
5
Views
858
  • Calculus and Beyond Homework Help
Replies
25
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
150
  • Calculus and Beyond Homework Help
Replies
1
Views
414
  • Calculus and Beyond Homework Help
Replies
2
Views
135
  • Calculus and Beyond Homework Help
Replies
1
Views
532
  • Calculus and Beyond Homework Help
Replies
5
Views
558
  • Calculus and Beyond Homework Help
Replies
9
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
465
Back
Top