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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?

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

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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 $A+B$?

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