M_n(R) matrix algebra

For each integer $n\geq1$ , $M_{n}(\mathbb R)$ consists of $n$ -by- $n$ matrices with entries in R, with entrywise addition and scalar multiplication and product

(XY)_{ij}=\sum_{r=1}^{n}X_{ir}Y_{rj}.

Its unit is $I_{n}$ . The scalar field for this catalogue object is $\mathbb R$ .

Dimension and multiplication

Each of the $n^2$ entries contributes $1$ real coordinates, so the real dimension is $n^2$ .

Matrix multiplication is associative because the coefficient algebra is associative; distributivity and rebracketing justify the equality of each entry of $(XY)Z$ and $X(YZ)$ .

Size convention

The entries for sizes one, two, and three are separate catalogue objects; this record is the parameterized family. Matrix size is not the dimension of the algebra.