Prove the following euqlaity of determinants

[Jennifer1990]

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

 

[tiny-tim]

Hi Jennifer1990!

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

Sooo … what is the definition of the determinant?

 

[Jennifer1990]

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

 

[Mark44]

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?

 

[Jennifer1990]

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

 

[Mark44]

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 A_ijs 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...

 

[Jennifer1990]

um actually, that's wat i meant o.o

 

[Mark44]

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

 

[Jennifer1990]

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?

 

[Mark44]

Yes, of course.

 

[tiny-tim]

(just got up :zzz:)

me too! ​

 

[Jennifer1990]

Is there another way to prove this question other than finding the determinant for the matrices?

 

[tiny-tim]

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 …

get on with it! ​