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Say, for example I have a 5 x 5 matrix, but every row and every column is filled with a zero. Is it still a 5x5 matrix or is it a 1 x 1 matrix with a zero?

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    $\begingroup$ I would just call it a zero matrix with dimensions as needed to suit your needs. $\endgroup$ – clocktower Apr 14 '16 at 0:35
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    $\begingroup$ It is still a $5$ by $5$ matrix. You can think of it as a linear transformation from $\mathbb{R}^5$ to itself. $\endgroup$ – fred Apr 14 '16 at 0:36
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    $\begingroup$ Yes, why would the values of the entries change the size? $\endgroup$ – user296602 Apr 14 '16 at 0:36
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It is still a $5 \times 5$ matrix. A matrix is essentially made up of three things:

1) A height (number of rows) $n$

2) A width (number of columns) $m$

3) A collection of entries (from some field) to fill up the $n \times m$ spaces.

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Yes it doesn't matter if it is filled with zeros or not, if it has five rows and five columns all zero it is still a $5\times 5$ matrix

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If I write $$ 0 = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{pmatrix} $$ it means that the zero matrix, here simply written as $0$. is the $5\times 5$ matrix with all elements being zero $0 \in \mathbb{F}$. So it is clear one works in $\mathbb{F}^{5\times 5}$ for some field $\mathbb{F}$.

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