TITLE: Why is that an $n \times n$ matrix have $n$ eigenvalues?
QUESTION [6 upvotes]: This is a silly question, but I can't find a good argument why an $n \times n$ matrix will have $n$ eigenvalues (not necessarily distinct).
In my linear algebra class the professor takes for granted that the number of eigenvalues is equal to the dimension of the matrix. For example, given eigs($A$) = $\{1, 2, 3, 4, 5\}$, he will write a $5 \times 5$ dimensional matrix for $A$
How can we show this trivial fact?

REPLY [2 votes]: Affirmative if you are taking matrices over the complex numbers, although not necessarily distinct. 
Over the real numbers the answer is sometimes a natural number $k$ between $0$ and $n$