Should we always regard a $1\times 1$ matrix as a scalar? Should we always regard a $1\times 1$ matrix as a scalar? (I think, "yes".) And if so, how should we address this in our elementaty Linear Algebra courses?
Let me give an example to illustrate my question.
Suppose $A= \left[\begin{array}{rr}1 & 2 \\-2 & 1\end{array}\right],$ $B = \bigl[-2\,,2\bigr],$ and
and $C = \left[\begin{array}{r}4 \\5\end{array}\right].$ Then the calculation
\begin{equation}  A(BC) = \left[\begin{array}{rr}1 & 2 \\-2 & 1\end{array}\right]
  \left(\begin{array}{r} \bigl[-2\,,2\bigr] \\ \rule{1pt}{0pt} \end{array}\left[\begin{array}{r}4 \\5\end{array}\right] \right) 
   = \left[\begin{array}{rr}1 & 2 \\-2 & 1\end{array}\right]\cdot 2 = \left[\begin{array}{rr}2 & 4 \\-4 & 2\end{array}\right]
\end{equation}
seems completely reasonable, doesn't it? And yet it's technically incorrect, since $A$ is a $2\times 2$ matrix and
$BC$ is a $1\times 1$ matrix.
I've ben trying to come up with a technically correct way to conclude that $A(BC)$ can indeed by computed as above. And here's the best I can
come up with. There's an obvious bijection, let's call it $J$, from the the $1\times 1$ matrices to the scalars, with $x = J([x])$ for
any scalar $x$. If we want to be able to carry out ``$A(BC)$'' as above, what we really mean is that it is equal to $AJ\bigl([BC]).$
But someone could ask how we know when it's appropriate to interpret $A(BC)$ as $2A$ and when it's
appropriate to interpret that product as undefined. My own answer is that it depends on context or something, but that seems unsatifying to me.
Does anyone know of a good way to address this matter, which is both rigorous at the foundational level and can easily be inserted into an
elementary discussion? For example, when we define multiplication of two matrices, should we add a caveat that any $1\times 1$ matrix should be
be regard as a scalar? But then, is there ever a situation where we want to regard a $1\times 1$ matrix as just that, and calling it scalar would
mess something else up at the level of foundations/definition?
Thanks in advance. -JGW
 A: While we represent both multiplication of a matrix by a scalar and multiplication of two matrices by juxtaposition of their multiplicands, they are actually different types of multiplication with different definitions (except in some special cases).  In my opinion, the difficulties highlighted in your question are caused by conflating these two different types of multiplication, rather than conflating scalars with $1\times1$ matrices.
To illustrate, let's use different symbols for the two types of multiplication, $*\ $  for multiplication of a matrix by a scalar, and $\ \circ\ $ for multiplication of two matrices.  The product $\ A*B\ $ is well-defined if and only if one or both of $\ A\ $ and $\ B\ $ are scalars (or $1\times1$ matrices) and the other is a matrix of any dimensions whatever.  The matrix product $\ A\circ B\ $, on the other hand, is well-defined whenever the number of columns in the matrix $\ A\ $ is the same as the number of rows in $\ B\ $.
If $\ A\ $ is a scalar (or a $1\times1$ matrix) and $\ B\ $ a row vector, or $\ B\ $ is a scalar (or a $1\times1$ matrix) and  $\ A\ $ a column vector, then both $\ A*B\ $ and $\ A\circ B\ $ are well-defined and they are equal. These are the only cases where both $\ A*B\ $ and $\ A\circ B\ $ are well-defined.
In your example, if you take $\ A(BC)\ $ to represent the product $\ A*(B\circ C)\ $, then it is well-defined and you can evaluate it exactly as you have done in your question.  Neither of the matrix products $\ A\circ(B\circ C)\ $ or $\ A\circ[2]\ $, however, is well-defined because the multiplicands have incompatible sizes.
A: Hint: I don't think it's appropriate to mix up the concept of matrices and scalars.
In linear algebra we have the fundamental relationship beween matrices and linear transformations. Multiplication of matrices and composition of linear transformations go hand in hand.

P. R. Halmos writes in Finite-Dimenstional Vector Spaces:

*

*Section 38: The relation between transformations and matrices is exactly the same as the relation between vectors and their coordinates,  ...


A: I would wager that your $BC$ is "really" an inner product of vectors, which (by convention) is a scalar. And then you're scalar-multiplying $A$ by that scalar.
So, no compulsion to think that a one-by-one matrix is identified with the corresponding scalar.
Some clarity may be achieved by writing such linear algebra stuff without reference to any choice of basis (hence, not matrices, but linear maps and vectors...)  If it becomes impossible to express the issue in such terms, it strongly suggests that there was a "type error" in the original matrix version of the question.
On another hand, there are some identities among matrix expressions of different sizes, e.g., for row vector $v$, $vv^\top=\mathrm{tr}\,v^\top v$. Such things enter in fancier identities such as mentions in the questioner's comment. But these do avoid "type errors", and have intrinsic expressions.
While we're here, we can debunk the somewhat-popular notion that "trace" of a square matrix has no intrinsic expression: first, for a finite-dimensional $k$-vectorspace $V$, the map $V\otimes V^*\to \mathrm{End}_k(V)$ by $(v\otimes\lambda)(w)=\lambda(w)\,v$ is an isomorphism. Then characterize trace by $\mathrm{tr}(v\otimes \lambda)=\lambda(v)$. :)
A: Matrices are different mathematical objects than scalars. There is a set of formal properties that must hold for them to be considered matrices.
A $1\times 1$ matrix made of a single element $x \in \mathbb{R}$ could be considered to be a scalar if and only if all properties that hold for the scalar $x$ hold for the $ 1 \times 1$ matrix made of $x$ without contradicting the properties of matrices and scalars.
Let $\begin{bmatrix} x\end{bmatrix}$ be the $1 \times 1$ matrix made of $x$. Assume $\begin{bmatrix} x\end{bmatrix} = x$. Then $\begin{bmatrix} x\end{bmatrix} + \begin{bmatrix} x\end{bmatrix} = x + x = 2x$, which is a real number. But matrix addition is defined as a closed operation; i.e., by definition the sum of matrices must return a matrix. Therefore the result contradicts the properties of matrices.
In conclusion, it is formally incorrect to treat a $1 \times 1$ matrix as a vector. I do not know if this is done in practice. However, I do not see why it should be done. Formal definitions serve a purpose, and not everything that seems intuitive must (or can) be.
