# For what real number $r$ is the set $\left.\left\{\begin{pmatrix}a & b \\ c & d\end{pmatrix}\right| a+b+c+d=r\right\}$ a subspace of $M_2$?

$M_2$ is the vector space of all $2\times 2$ matrices with real entries. For what real number r is the set $$\left.\left\{\begin{pmatrix}a & b \\ c & d\end{pmatrix}\right|a+b+c+d=r\right\}$$ a subspace of $M_2$?

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What is one vector that every subspace must contain? – wj32 Oct 21 '12 at 7:38
You mean 0 vector? – John Hass Oct 21 '12 at 7:45
Yes. You should be able to do the question now. – wj32 Oct 21 '12 at 7:47
Thank you @wj32 – John Hass Oct 21 '12 at 7:50
Peng, now that you can answer the question, you should post an answer. Later, you can accept it. – Gerry Myerson Oct 21 '12 at 11:39

1. A quick solution:

Every subspace must contain zero vector, which means $\left\{\begin{pmatrix}0 & 0 \\ 0 & 0\end{pmatrix}|0+0+0+0=r\right\}$ is in the subspace of $M_{2}$. Hence, $r$=0.

2. Alternative solution:

We can use the definition of subspace to find $r$ : closed under addition and scalar multiplication.

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