Please explain what is meant by "defined". I have a MCQ type question that goes like this:
If $A$ is an $m \times n$ matrix, $B$ is an $n \times p$ matrix and $C$ is a $p \times n$ matrix, then
a) $AB$ is defined for all $m$,$n$,$p$.
b) $A^TC$ is defined for all $m$,$n$,$p$.
c) $BC$ and $CB$ are both squre matrices.
d) $B^TC^T$ is of order $n \times n$.
e) none of above.
I know that the answer c) is correct and d) is wrong.
But I do not understand what is meant by answers a) and b). Please help.
By "defined", it means that the resulting matrix exists. For example you should know that if you have matrix $A$ that is $2 \times 4$ and matrix $B$ that is $3 \times 5$, the product of these matrices $AB$ doesn't exists because the dimensions don't match. If this happens, we can say that $AB$ is "not defined". The other product $BA$ is also "not defined" for the same reason. The matrix product $AB$ is only "defined" if the number of columns in $A$ is the same as the number of rows in $B$ (the dimensions match).