Multiplying by a $1\times 1$ matrix?

For matrix multiplication to work, you have to multiply an $m \times n$ matrix by an $n \times p$ matrix, so we have $$\bigg(m \times n\bigg)\bigg( n\times p \bigg).$$

But what about a $1 \times 1$ matrix? Is this just a scalar? But every matrix can be mulitplied by a scalar; so do $1 \times 1$ matrices break the rule?

Solution 1:

A $1\times 1$ matrix is not the same thing as a scalar. There is a one to one correspondence between them, but $[a]$ and $a$ are two different things, the first being a matrix and the second one being a scalar. The confusion comes from the fact that the rings $R$ and $M_1(R)$ are isomorphic, and so $M_n(R)$ is isomorphic to $M_n(M_1(R))$. When you talk about treating $[a]$ as a scalar, you are implicitly using this isomorphism.

Solution 2:

Matrix multiplication can be regarded as a generalization of the vector product.

An $n$-dimensional row vector can be multiplied by an $n$-dimensional column vector. The result is a scalar value known as the dot product (a special case of the inner product).

These two vectors can also be regarded as matrices: an $1\times n$ multiplied by $n\times 1$. The result should then be a $1\times 1$ matrix, and the element of that matrix will be that dot product.

There is certainly room for regarding $1\times 1$ matrices as scalars, when doing so is convenient.

There is no conflict between the product of a matrix by a scalar, and the product of two $1\times 1$ matrices. For instance $2 \times [3] = [6]$ and also $[2][3] = [6]$.

Of course the ${scalar}\times {matrix}$ case is not restricted by $m\times n/n\times k$ compatibility: it just scales every matrix alement by the scalar. But that semantic difference vanishes in the $1\times 1$ case.

So no rule is really broken! In all cases where you can multiply a matrix by either a scalar or a matrix containing just that scalar, the result is the same.

But scalar multiplication has an additional freedom. Typographically speaking, if you "drop the square brackets" from the $1\times 1$ matrix, you gain the flexibility of doing a different kind of multiplication.