Does this addition of matrices work?

Question

Can you do scalar plus a matrice? Such as $2+E$ ($2$ being the scalar and $E$ a matrix) and if you can't why?

Answer 1

Note that if you want the addition to work like $$2 + \begin{pmatrix} a & b\\ c & d \end{pmatrix} = \begin{pmatrix} 2+a & 2+b\\ 2+c & 2+d \end{pmatrix}$$ then what you are really doing is $$2\begin{pmatrix} 1 & 1\\1 & 1 \end{pmatrix} + \begin{pmatrix} a & b\\ c & d \end{pmatrix} = \begin{pmatrix} 2+a & 2+b\\ 2+c & 2+d \end{pmatrix}.$$ So no, it's not a good idea to write a scalar plus a matrix because it is ambiguous, but there are simple ways to write what you are trying to accomplish using proper notation.

Answer 2

In general: No you cannot, because dimension. What is $$2+\begin{bmatrix} a & b\\ c & d\end{bmatrix}$$ supposed to mean? $$\begin{bmatrix}a+ 2 & b \\ c & d\end{bmatrix}? \;\; \begin{bmatrix}a+2&b+2\\c+2&d+2\end{bmatrix}? \;\; \begin{bmatrix}a+2&b\\c&d+2\end{bmatrix}?$$

What I want to say is: There is no obvious choice, what scalar + matrix is supposed to mean. If you want to use such notation, you will have to make clear, what kind of choice you made.

Answer 3

What you want is a map $k\times M_n(k) \to M_n(k)$. Scalar multiplication is such a map, for example. This wouldn't be considered an operation, but an action. Anyway, not to go into too much details, there is obviously many ways to do it, but you would probably want some axioms to hold as well.

One natural way to do it is pick any matrix $E\in M_n(k)$ and define $$\alpha + A := \alpha E + A$$ where the right hand side operations are the standard ones. Notice that $$(\alpha + \beta) + A = (\alpha + \beta)E + A = \alpha E + (\beta E + A) = \alpha + (\beta + A),\\ 0 + A = 0\cdot E + A = A,$$ so this defines an action of the additive group $(k,+)$.