Which of the following are binary operations? Please let me know if my answers a

Question

Which of the following are binary operations?

Please let me know if my answers are wrong and if they are why?

1) Taking an invers on the set of all two by two matrices: Yes

2) Addition on the set of two by two matrices with trace 0. (Recall that the trace of a matrix is the sum of the entries on the diagonal. For example the trace of (1 2)    is 1 + 8 = 9.): Yes

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3) Addition on the set of two by two matrices with trace 1: No

4) Multiplication on the set of two by two matrices with determinant 0: Yes

5) Addition on the set of two by two matrices with determinant 0: No

6) Multiplication on the set of two by two matrices with determinant 1: Yes

7) Multiplication on the set of two by two matrices with determinant 2: No

8) Taking an inverse on the set GL (2, R): Yes

9) Taking an inverse on the set GL (2, Z): Yes

10) Addition on the set of two by two matrices with exactly one nonzero entry: No

11) Multiplication on the set of two by two matrices with exactly one nonzero entry: No

Explanation / Answer

1) Taking an invers on the set of all two by two matrices: Yes

2) Addition on the set of two by two matrices with trace 0. (Recall that the trace of a matrix is the sum of the entries on the diagonal. For example the trace of (1 2)    is 1 + 8 = 9.): Yes

3) Addition on the set of two by two matrices with trace 1: No

4) Multiplication on the set of two by two matrices with determinant 0: Yes

5) Addition on the set of two by two matrices with determinant 0: No

6) Multiplication on the set of two by two matrices with determinant 1: Yes

7) Multiplication on the set of two by two matrices with determinant 2: No

8) Taking an inverse on the set GL (2, R): Yes

9) Taking an inverse on the set GL (2, Z): Yes

10) Addition on the set of two by two matrices with exactly one nonzero entry: No

11) Multiplication on the set of two by two matrices with exactly one nonzero entry: No

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