Some matrices are real numbers?

Question

Suppose $a\in \mathbb{R}$, $A$ is a matrix, and $I$ is the identity matrix. Then, $$ I a = I a \\ A = Ia\\ IA = IIa \\ IA=Ia\\ \implies A=a $$

Answer 1

If $M$ is an invertible matrix and $A$ and $B$ are matrices (all the matrices here are square matrices with the same size), then, yes, $MA=MB\implies A=B$. But in the equality $\operatorname{Id}A=\operatorname{Id}a$, $\operatorname{Id}a$ is not a product of two matrices. So, you cannot apply that property here.

Answer 2

The basic problem is that multiplication of a matrix by a scalar ($Ia$) and multiplication of matrices ($IA$) are really different operations although the notations look the same.

That said, the matrix $Ia$ behaves just like the scalar $a$ in some ways, and the two are often identified. Thus $(Ia)M = a M = M (Ia)$ if both $M$ and $I$ are $n \times n$ matrices.