When the modeling matrix is on the left and the algebraic worker is on the right, we chop the data into columns.

1. Start with modeling matrix $A$ in the left argument
2. Strategically choose algebraic worker $X$
3. Hit $A$ on the right with $X$ (place $X$ in right argument)
4. Produce the output product $B$

• ! The inner dimensions must agree!
  • Number of rows of $A$ must equal number of columns of $X$
  • If dimensions agree, $A$ is conformable by $X$ for right matrix-matrix multiplication

$$\underset{\text{algebraic worker}}{\stackrel{\text{left factor}}{X}}\cdot \underset{\text{ modeling matrix}}{\stackrel{\text{right factor}}{A}}=\stackrel{\text{product}}{B}$$

1. Start with modeling matrix $A$ in right argument
2. Strategically choose algebraic worker $X$
3. Hit $A$ on the left with $X$ (place $X$ in the left argument)
4. Produce the output product $B$

• ! The inner dimensions must agree!
  • Number of rows of $X$ must equal number of columns of $A$
  • If dimensions agree, $A$ is conformable to $X$ for left matrix-matrix multiplication

We have two sides because unlike scalar multiplication, matrix-matrix multiplication is not exactly commutative. The order matters because matrices have dimensions.

• Right matrix-matrix multiplication is for working with columns
• Left matrix-matrix multiplication is for working with rows