Multiplying matrices general question If I have a matrix with the dimensions $2\times3$ and another matrix with the dimension of $2\times4$, I learned that I would not be able to multiply it since the inner dimensions need to match meaning the column from the first matrix needs to match the row of the second matrix.
My professor said that I am able to add a row of $1's$ to the second matrix and would have the dimension $3\times4$ and then I can multiply.
I am not understanding why I am allowed to do this. Also, why $1$'s and not $0$'s? I am guessing it has something to do with the identity matrix.
 A: The left matrix must have as many columns as the right matrix has rows as the value for the elements in the product are pair-wise scalar products of rows from the left and columns from the right.
Usually one writes matrix dimensions as $Rows \times Columns$. So if you write the left matrix dimensions $R_L \times C_L$ and right as $R_R \times C_R$ next to each other like this: $$R_L \times C_L , R_R \times C_R$$The inner dimensions are in the middle ($C_L$ and $R_R$, right). Those need to be the same and it is those who are "eaten up" when the matrix multiplication takes place. So we go from 4 dimensions down to 2 and the result matrix will have the "outer" dimensions $$R_L \times C_R$$
Usually a matrix is required to have the same type of elements everywhere so we know how to do addition and multiplication with them. What you add to the new row needs to be from the same field of numbers.  A field is basically a bunch of numbers together with the rules of how to do multiplication and addition with them. But these rules include that there must always exist 0 and 1 for any field so you can always add zeros if you want to. But maybe sometimes it won't be useful for what you are trying to build.
