# Set of a matrix

I am working on a homework problem which asks me about the Set of a singular $n\times n$ matrix. specifically whether it is a vector space. I looked in the glossary of the book and searched online and no where could I find a definition of the Set of a matrix. What is the definition of a matrix set? Is it just the column space?

• Presumably the problem means (for a given $n$) the set whose elements are the singular $n \times n$ matrices. Commented Jun 9, 2015 at 11:57
• oh you are exactly correct, I misread the question. Thank you. Commented Jun 9, 2015 at 11:59

So you're studying the set $S \subset \mathcal{M}_n(\mathbb{R})$ of singular matrices.

The matrices:

$$A=\left( \begin{array}{cccc} 1 & 0 & \dots & 0 \\ 0 & 0 & \dots & 0 \\ \vdots & & \ddots & \\ 0 & & & 0\\ \end{array} \right)$$ and

$$B=\left( \begin{array}{cccc} 0 & 0 & \dots & 0 \\ 0 & 1 & \dots & 0 \\ \vdots & & \ddots & \\ 0 & & & 1\\ \end{array} \right)$$

are singular (elements of $S$). What do you think of $A+B$?