Turn on thread page Beta
    • Thread Starter
    Offline

    1
    ReputationRep:
    Over a field F, you can invert a matrix if det  \not= 0.

    Is this also true for a field. Also if det=0 are there still cases where you can invert the matrix, or is this mean to be:


    Over a field F, you can invert a matrix IFF det  \not= 0??

    Cheers.
    Offline

    10
    ReputationRep:
    I'm a bit rusty on this but reckon it's the IFF statement.

    If you want to generalise, a matrix over a commutative ring is invertible IFF the determinant is a unit in that ring.

    Anyone back me up?
    • Thread Starter
    Offline

    1
    ReputationRep:
    (Original post by vc94)
    If you want to generalise, a matrix over a commutative ring is invertible IFF the determinant is a unit in that ring.
    I have definitely seen this in my notes today, can anyone confirm that it is the IFF statement??

    Cheers
    Offline

    13
    ReputationRep:
    (Original post by adie_raz)
    I have definitely seen this in my notes today, can anyone confirm that it is the IFF statement??

    Cheers
    If A is invertible, there is a matrix B such that AB=I. Then det(A)det(B) = 1. So det(A) can't be zero.
    Offline

    10
    ReputationRep:
    (Original post by vc94)
    I'm a bit rusty on this but reckon it's the IFF statement.

    If you want to generalise, a matrix over a commutative ring is invertible IFF the determinant is a unit in that ring.

    Anyone back me up?
    Yes, that's correct. To be even more general, there is a matrix adj(A) such that adj(A) A = det(A) I.
    • Thread Starter
    Offline

    1
    ReputationRep:
    (Original post by Zhen Lin)
    Yes, that's correct. To be even more general, there is a matrix adj(A) such that adj(A) A = det(A) I.
    Since it is correct, can you quickly explain how to show if the determinant is a UNIT in the ring, or will this take a while, and therefore need a fair bit of work??

    if so I will try and revise it and come back with more specific questions. Thanks
    Offline

    13
    ReputationRep:
    (Original post by adie_raz)
    Since it is correct, can you quickly explain how to show if the determinant is a UNIT in the ring, or will this take a while, and therefore need a fair bit of work??

    if so I will try and revise it and come back with more specific questions. Thanks
    If AB=I then det(A)det(B)=1 so det(B) is the multiplicative inverse of det(A). So det(A) is a unit.
    • Thread Starter
    Offline

    1
    ReputationRep:
    (Original post by SsEe)
    If AB=I then det(A)det(B)=1 so det(B) is the multiplicative inverse of det(A). So det(A) is a unit.
    Does this also mean that det(B) is a unit then, by the same argument.
    Offline

    12
    ReputationRep:
    Yes
    • Thread Starter
    Offline

    1
    ReputationRep:
    Thanks
 
 
 
Reply
Submit reply
Turn on thread page Beta
Updated: March 24, 2011
The home of Results and Clearing

2,797

people online now

1,567,000

students helped last year

University open days

  1. SAE Institute
    Animation, Audio, Film, Games, Music, Business, Web Further education
    Thu, 16 Aug '18
  2. Bournemouth University
    Clearing Open Day Undergraduate
    Fri, 17 Aug '18
  3. University of Bolton
    Undergraduate Open Day Undergraduate
    Fri, 17 Aug '18
Poll
Do you want your parents to be with you when you collect your A-level results?
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

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

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