You are Here: Home >< Maths

# Proving determinant of a matrix is equal to product of eigenvalues Watch

Announcements
1. Hey guys, well that's the question above ^

det(AB)=det(A)det(B)
and det(B^-1)=1/det(B)

SO
det(D)=det(P^(-1)AP)
=det(P^-1)det(AP)
=det(P^-1)det(A)det(P)
=(1/det(P))det(A)det(P)
=det(A)

Which leaves us with det(A)=det(D)

Am I going in the right direction for a start to prove this?
And if I am any advice (just advice rather than answers is probably best ) on how to go from here?

Thanks
2. Yeah. What is D? And what is its determinant?
3. (Original post by SimonM)
Yeah. What is D? And what is its determinant?
So, D is a diagonal matrix containing the eigenvalues of A on it's diagonal, therefore I guess it's determinant is indeed the product of those values. However, how could I show this; guessing it isn't enough to just state it..
4. I would just state it. If you absolutely had to prove it, use cofactor expansion and induction on the size of the matrix.
Hey guys, well that's the question above ^

det(AB)=det(A)det(B)
and det(B^-1)=1/det(B)

SO
det(D)=det(P^(-1)AP)
=det(P^-1)det(AP)
=det(P^-1)det(A)det(P)
=(1/det(P))det(A)det(P)
=det(A)

Which leaves us with det(A)=det(D)

Am I going in the right direction for a start to prove this?
And if I am any advice (just advice rather than answers is probably best ) on how to go from here?

Thanks
since you have found det(A)=dat(D), nd D is diagonal matrix and you know the entries of the diagonal matrix are e.values of A ,and its determinant is product of diagonal entry. U r done !!!

but , here is a problem , as every square matrix A is not diagonalizable so u can't guarantee the existence of such D always,in case for A there is no such D (diagonal matrix) then ??

i think it would be easier to think in terms of Characteristic polynomial of A .

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: March 20, 2013
Today on TSR

### Things NOT to do at uni

Have you done any of these?

### How much revision have yr12s done?

Discussions on TSR

• Latest
• ## See more of what you like on The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

• Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams

## Groups associated with this forum:

View associated groups
Discussions on TSR

• Latest
• ## See more of what you like on The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

• The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.