(a) Show that H is closed under matrix multiplication.
(b) The identity matrix is not an element of H. Nevertheless, H has an identity element (using the operation of matrix multiplication). Find it, and verify that the identity axiom holds.
(c) Using the identity element you found in (b), show that every element of H has an inverse (using the operation of matrix multiplication).
This problem shows that a set with a familiar operation may have an identity and inverses which are not the usual ones.
Hence, is the identity for H under matrix multiplication.
Thus, is the inverse of in H.
2. is a group under component-wise addition:
(a) Verify that satisfies the identity axiom for this operation.
(b) Verify that if , then the inverse of is .
(c) Show that the following subset of is a subgroup of :
(d) Show that the following subset of is not a subgroup of by finding one subgroup axiom that is violated:
(c) If , then
If , then
Therefore, H is a subgroup of .
(d) The identity is , and .
If , the inverse is , and .
However, and . But : Since , it follows that can't be written as for any integer y. Thus, K is not closed under the operation, and it isn't a subgroup of .
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