subsets of M3R

Let G be a subset of Msub3(R) satisfying the following three conditions:

1. For every A and B in G, the matrix product AB is in G
2. There is a matrix E in G such that EA = A = AE for all A in G
3. For every A in G, there exists a B in G such that AB = E = BA

I'm supposed to figure out if the following statements are true or false. Prove our counterexample them.

1. E is the 3 x 3 identity matrix.
2. If A is in G then Det(A) is not equal to zero.

My feeling is that 1 is true, but I don't know how to prove it. And I'm not sure about 2.

For Q1, try multiplying your matrix E by the identity.

For Q2, take the determinant of the product of matrices in property 3. Suppose the determinant of A were 0. Is this possible?

Quote:

---

Originally Posted by grandunification

Let G be a subset of Msub3(R) satisfying the following three conditions:

1. For every A and B in G, the matrix product AB is in G
2. There is a matrix E in G such that EA = A = AE for all A in G
3. For every A in G, there exists a B in G such that AB = E = BA

I'm supposed to figure out if the following statements are true or false. Prove our counterexample them.

1. E is the 3 x 3 identity matrix.
2. If A is in G then Det(A) is not equal to zero.

My feeling is that 1 is true, but I don't know how to prove it. And I'm not sure about 2.

---

The set $\displaystyle G=\{0\} \in\,M_3(\mathbb{R})$ fulfills the conditions and (1)-(2) are false.

Tonio

Ya, that's what I was gonna due, but I doubted myself.

There's still a slight problem though: namely that you want me to take EA = A = AE for all A in G, and use I for A. But how do we know that I is in G?

I guess you could say:
1. Let B be in G then IB = B which is in G by assumption.
2. EI = I = IE for E = I.
3. (I)(I) = I = (I)(I)