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?
2026-04-29 16:15:50.1777479350
Set of a matrix
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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 $A+B$?