Scalars matrices is a normal subgroup of $GL_2(\mathbb{R})$

Question

Prove or disprove the following assertion.

The set of all nonzero scalars matrices is a normal subgroup of $GL_2(\mathbb{R})$.

Proof:

Let $I$ be the identity matrix. Consider the scalar matrix $sI$ where $s$ is some scalar.
Then let $A$ be any other matrix in $GL_2(\mathbb{R})$.
So $A(sI)A^{-1} = sAIA^{-1} = sAA^{-1} = s = sI$.
Then if $H$ is the subgroup of scalar matrices, then $AHA^{-1} = H$. Thus $H$ is normal.
Hence, $H$ is the center of $GL_2(\mathbb{R})$.

My professor wrote on my homework to show it is a subgroup. I am not sure how to do that.

Answer 1

Hint: show that the product of any two scalar matrices is a scalar matrix, and that the inverse of a scalar matrix is another scalar matrix.

Answer 2

Note that scalar matrices commute with all other matrices, so if we can show that such matrices form a subgroup, then it is automatically a normal subgroup. It looks like you've gotten this far already.

First, note that the identity is a scalar matrix. Next, we want to show the following:

1. The inverse of a scalar matrix is a scalar matrix.
2. The product of two scalar matrices is again a scalar matrix.

These are both pretty easy to show, so I'll leave that work to you. Once you have these facts, then you can conclude that it is indeed a subgroup.