So I have two matrices:
[2x3] x [2x5]
At their current state, these two matrices are unable to be multiplied. Right? Would I transpose the [2x3] matrix into a [3x2] so I can multiply it with the second one? Am I able to use the transpose operation in a problem like this?
Thanks!
Short answer: if they ask you to find the product of one 2x3 dimensional matrix and one 2x5 matrix, the answer is undefined, and you need to explain why this is the case, how is that the definition of matrix multiplication doesn't include this instance.
Long answer: Matrix multiplication has a motivation. Transposition has a motivation too, albeit a little more sophisticated. I will tell you about multiplication only: a matrix can store coordinate transformations of vector components, in a way that to apply the transformation to the vector is the same as taking the product of the matrix with the column matrix of the components of said vector, obtaining a new, visually distorted vector, possibly in a different enviroment from the one of the initial vector (when a spatial vector is sent to a planar vector, for example). This is a geometrical interpretation, which is not essential to the matrix algebra and its general dimensions, but it can give you a picture when thinking of 2x2, 2x1 and 2x3 matrices. A matrix times other matrix represents the composition of geometrical transformations. In this respect, a 2x3 matricx and a 2x5 are transformations of diferent kinds of vectors, in this case 3-dimensional and 5-dimensional vectors, respectively. Therefore, the transposition of one of them loses its original meaning, and you would need to have a stronger justification for fitting this 2x3 and 2x5 matrices into a 3x2 times 2x5 product matrix (a justification that may involve other new concepts such as dual space), otherwise you are doing a mechanically correct but meaningless procedure.