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Prove that this is a field...

Let F be the set of all 2x2 real matrices of the form

A=\begin{bmatrix} a & b \\-b & a \end{bmatrix}

(i) Prove that F is a field (under the operation of addition and multiplication).

For multiplication:

We know that the identity is not equal to zero, because...

[1001]≠[0000]\begin{bmatrix} 1 & 0 \\0 & 1 \end{bmatrix} \not= \begin{bmatrix} 0 & 0 \\0 & 0 \end{bmatrix}

But for the inverses...we know that every matrix in A has an in verse if the determinant is not zero...the determinant is a^2 + b^2. So the determinant will only be zero if both a and b are zero, but then we will have the zero matrix...which doesn't need to have an inverse...

For addition:

I'm a bit confused because we know that the identity matrix for addition is the zero matrix. So the identity is equal to zero...then how can if be a field?

Thanks in advance
Original post by Artus

Thanks in advance


In a field there is no "the identity". There is an additive identity, and a multiplicative identity.

Does that answer your question, or are you asking something else?
(edited 11 years ago)
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Artus
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Original post by ghostwalker
In a field there is no "the identity". There is an additive identity, and a multiplicative identity.

Does that answer your question, or are you asking something else?


Thanks, but in our textbook, it says that 1≠01 \not= 0 in a field...so does that mean that both the additive and the multiplicative identity should not be equal to zero? Because in this question, we know that the additive idenity is the zero matrix, so how can it be a field?
Original post by Artus
Thanks, but in our textbook, it says that 1≠01 \not= 0 in a field...so does that mean that both the additive and the multiplicative identity should not be equal to zero? Because in this question, we know that the additive idenity is the zero matrix, so how can it be a field?


1≠01 \not= 0 means the multiplicative identity does not equal the additive identity.

0 is, by definition, the additive identity.
And similarly 1, is by definition, the multiplicative identity.

They're standard ways to represent the identities.

In this case 1 is the identity matrix, and 0 is the matrix of all zeroes.
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Artus
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Original post by ghostwalker
1≠01 \not= 0 means the multiplicative identity does not equal the additive identity.

0 is, by definition, the additive identity.
And similarly 1, is by definition, the multiplicative identity.

They're standard ways to represent the identities.

In this case 1 is the identity matrix, and 0 is the matrix of all zeroes.


Ok, thanks. :smile:

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