is a 1x1 matrix the same as a scalar?

Question

is a 1x1 matrix the same as a scalar?

Answer 1

Technically speaking, no.



Remember, when multiplying matrices, their sizes need to satisfy certain conditions.

That is, if A is m x n and B is p x q then the product AB is only defined when n=p and the resulting matrix is size m x q.



So when you look at multiplication by a scalar c to any matrix A of size m x n, the scalar product is defined. However, if you consider the 1x1 matrix consisting of c and any matrix A of size m x n, then the product cA is not necessarily defined, unless of course, m=1.



Edit: When you used the word "scalar" I took it to mean an element from your field used to /scale/ vectors from your vector space. Reread what I wrote with that in mind. If you want a better answer, ask a better question.

Answer 2

A scalar quantity can have at most one single dimension. It MAY have a direction ( forward and backwards) but no more than that. So use these rules. Can time have more than one dimension or just a single one? Can time have a value in either the forward or backward direction but nothing other than this? If so then it may be considered to be a scalar. A vector has multiple independent dimensions where there are many conditions in which 1+1 does NOT equal 2 ( nor zero either). i.e 1 North + 1 east does not equal 2 in any direction. Only if we artificially constrain the movement along a single track ( railway track?) then the movement can be considered as a scalar.

Answer 3

In most practical situations you can think of a 1x1 matrix as a regular "scalar" number.



But Makar is likely correct. Technically they may be different, but in practice if you get an answer that is a 1x1 matrix you can usually convert it to a single number and work on that way.