M_3(C) matrix algebra

$M_{3}(\mathbb C)$ consists of $3$ -by- $3$ matrices with entries in C, with entrywise addition and scalar multiplication and product

(XY)_{ij}=\sum_{r=1}^{3}X_{ir}Y_{rj}.

Its unit is $I_{3}$ . The scalar field for this catalogue object is $\mathbb C$ .

Dimension and multiplication

The real dimension is $18$ , since there are $9$ entries with $2$ real coordinates each. Its complex dimension is $9$ .

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

For every size at least two, $E_{11}E_{22}=0$ exhibits nonzero zero divisors. Matrix size is not the dimension of the whole algebra.