# Determine whether the given set is closed under the usual addition and scalar multiplication, and is a (real) vector space [closed]

The set of all 2 x 2 matrices of the form

\begin{bmatrix} x & 1 \\ 1 & x \end{bmatrix}

Where each $x$ may be any scalar

I don't get why this doesn't close under addition textbook says thats the reason this isn't a vector space.

## closed as off-topic by Hurkyl, Mercy King, Claude Leibovici, Siong Thye Goh, k1.MSep 10 '17 at 14:31

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• Try adding two of them. Any two of them, it doesn't matter which. – Hurkyl Sep 10 '17 at 3:02
• Incidentally, the body of your post should be self-contained -- it's okay to duplicate information in the title, but nothing important should be only in the title. – Hurkyl Sep 10 '17 at 3:03

Hint: $$\begin{bmatrix}x & 1\\1&x\end{bmatrix} + \begin{bmatrix}y & 1\\1&y\end{bmatrix}=\begin{bmatrix}x+y & \color{red}{2}\\\color{red}{2}&x+y\end{bmatrix}.$$
• @jimmyjimmy Let the set be $S$ whose elements are the matrices of the given form. The set is closed under addition if the sum of any two elements in $S$ results in another element in $S$. Does $\begin{bmatrix}x+y & \color{red}{2}\\\color{red}{2}&x+y\end{bmatrix}$ belong to $S$? – Math Lover Sep 10 '17 at 3:09