Matrix Algebra: Operations, Rules, and the Transpose

The previous note treated elimination steps as matrix multiplications without pinning down the rules that make multiplication well defined. This note supplies them — addition, scalar multiplication, matrix multiplication, and the transpose — and flags the single most consequential fact a newcomer to matrix algebra must internalise: matrix multiplication does not commute. That asymmetry is not a technicality; it is the algebraic shadow of the fact that "apply hedge $A$, then rebalance $B$" is a genuinely different operation from "rebalance $B$, then apply hedge $A$." The transpose closes the note by building the object — $A^\top A$ — that underlies every covariance matrix in the unit.

Learning objectives. By the end of this note you should be able to: