Question

Determine if the subset of Mn,n is a subspace of Mn,n, with the standard operations: The set of all nxn matrices with integer entries.

Answer 1

Let \(A\) be the subset in questions, and let \(X,Y\in A\). Such a matrix will look like this:
\[X=\begin{bmatrix}x_{1,1}&x_{1,2}&\cdots&x_{1,n}\\
x_{2,1}&x_{2,2}&\cdots&x_{2,n}\\
\vdots&\vdots&\ddots&\vdots\\
x_{n,1}&x_{n,2}&\cdots&x_{n,n}\end{bmatrix}~~~~Y=\begin{bmatrix}y_{1,1}&y_{1,2}&\cdots&y_{1,n}\\
y_{2,1}&y_{2,2}&\cdots&y_{2,n}\\
\vdots&\vdots&\ddots&\vdots\\
y_{n,1}&y_{n,2}&\cdots&y_{n,n}\end{bmatrix}\]
where \(x_{i,j},~y_{i,j}\in\mathbb{Z}\) for all \(i,j\in\{1,2,\cdots,n\}\).

In order for a subset to be a subspace, you must show that for any such \(X.Y\in A\) you have
(1) \(X+Y\in A\) and
(2) \(cX\in A\) for \(c\in\mathbb{R}\).

Answer 2

Thanks SithsandGiggles! So am I correct in that this subset would not qualify as a subspace because it fails to have closure under scalar multiplication?

Answer 3

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