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# Invertible Matrices watch

1. Over a field F, you can invert a matrix if det 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 0??

Cheers.
2. 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?
3. (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
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.
5. (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.
6. (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
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.
8. (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.
9. Yes
10. Thanks

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