Is it possible to have an $a \times b \times c$ matrix?

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The book Artificial Intelligence: A Modern Approach states that a certain variable is a $2 \times 2 \times 2$ matrix", but I thought that matrices could only be rectangular (i.e. $a \times b$). Is it possible to have a matrix that is $a \times b \times c$?

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Yes, there is such a generalisation. It's called tensor.