What are the dimensions of the product of two matrices? A simple question
is a (5x2)*(2x5) = a (5x5) matrix?
 A: Yes, It is indeed a $5 \times 5$ matrix. For any matrix multiplication to be defined for some arbitrary matrices $A$ and $B$, $A$ could be a $m \times n$ and $B$ could be a $p \times q$, in this order, for the matrix multiplication to be defined. The amount of columns of matrix $A$ must equal the amount of rows of matrix $B$, (i.e., $n=p$).
So a $~\Rightarrow~$ $m \times n$ $\cdot$ $p \times q$ will result in a $m \times q$ matrix $\iff~~n=p$
$\underline{\text{Example:}}~~$ $3 \times 4$ $\cdot$ $4 \times 3$ = $3 \times 3$ matrix.  
I hope that this helps out a bit.
Good~Luck.  
A: A matrix of order $m \times n$; where $m$ is the number of rows and $n$ the number of columns 'represents' a linear map that takes input from an $n-$ dimensional vector space and outputs in a $m-$ dimensional vector space. The rules of multiplying matrices comes from the fact that we want matrix multiplication to reflect the composition of two maps. Since, for composition of maps to work, the (space) for the output of the first map should coincide with the (space) of the input of the next map. Hence the rule that the column of one matrix should coincide with the row of the other.   
A: Yes..
the way that i learnt it, when a 5*2 matrix is mutiplied into a 2*5 matrix, the middle number get taken away, leaving only the outer numbers, so in this case, the twos get taken away, leaving only the fives..
