I came upon the fact that if you defined matrix multiplication such that $ \underbrace{\begin{pmatrix} a&b&c \ a’&b’&c’ \ a”&b”&c” \end{pmatrix}}_A \begin{pmatrix} x&y&z \end{pmatrix} = \begin{pmatrix} X&Y&Z \end{pmatrix}$

for $ X = ax + by +cz$ and so on, and then defined

$ \begin{pmatrix} x&y&z \end{pmatrix} = \underbrace{\begin{pmatrix} \alpha&\beta&\gamma \ \alpha’&\beta’&\gamma’ \ \alpha”&\beta”&\gamma” \end{pmatrix}}_B \begin{pmatrix} \delta&\epsilon&\sigma \end{pmatrix}$

you’ll end up with the fact that the matrix $ AB$ is the same as defined by the normal convention of multiplying matrices.

(Page 20, Arthur Cayley’s *A Memoir of the Theory of Matrices* http://scgroup.hpclab.ceid.upatras.gr/class/LAA/Cayley.pd)

So I tried doing the same process with two compound matrices (instead of one compound matrix and a row matrix as is done in the example above).

And I ran into a problem. If we defined a sort of $ \text{row} \cdot \text{row}$ multiplication such that the first Row of the first matrix multiplies with the rows of the second matrix to form the first row of the resultant matrix, the entire convention becomes inconsistent.

As you can see,

$ \begin{matrix} \tiny{R_1} \ \tiny{R_2} \end{matrix} \begin{pmatrix} a&b\a’&b’ \end{pmatrix} \cdot \begin{matrix} \tiny{r_1}\\tiny{r_2} \end{matrix} \begin{pmatrix} c&d\c’&d’ \end{pmatrix} = \begin{pmatrix} ac+bd & ac’+bd’\ a’c+b’d & a’c’+b’d’ \end{pmatrix}$

And then we defined the $ cd$ matrix as a product of two other matrices:

$ \begin{pmatrix} c&d\c’&d’ \end{pmatrix} = \begin{pmatrix} X&Y\X’&Y’ \end{pmatrix} \begin{pmatrix} x&y\x’&y’ \end{pmatrix}$

And then if we tried to find a $ lm$ matrix such that,

$ \begin{pmatrix} l&m\l’&m’ \end{pmatrix} \begin{pmatrix} x&y\x’&y’ \end{pmatrix} = \begin{pmatrix} a&b\a’&b’ \end{pmatrix} \begin{pmatrix} c&d\c’&d’ \end{pmatrix}$

You get this:

$ \begin{pmatrix} lx+my&lx’+my’\l’x+m’y&l’x’+ m’y’ \end{pmatrix} = \begin{pmatrix} aXx+aYy+bXx’ +bYy’ & aX’x+aY’y +bX’x’ +bY’y’\a’Xx+a’Yy+b’Xx’ +b’Yy’ & a’X’x+a’Y’y +b’X’x’ +b’Y’y’ \end{pmatrix} \begin{pmatrix} c&d\c’&d’ \end{pmatrix}$

Which, if I’m not wrong, is a nonsensical matrix equation since $ x’$ s and $ y’$ s appear in RHS in places where they can’t occur in LHS.

But I can’t find a satisfactory explanation of why that inconsistency occurs.

On the other hand, if I defined this “row $ \cdot$ row” matrix multiplication as $ r_1 \cdot (R_1 R_2 R_3)$ , i.e., the first row of the second matrix multiplied with the rows of the first matrix to form the first row of the resultant matrix, the method works as it should have. I end up at the matrix multiplication convention that is already in use.

And I can’t find a great insight to why $ r_1 \cdot (R_1 R_2 R_3)$ works but $ R_1 \cdot (r_1 r_2 r_3)$ , which looks more promising, gives inconsistent result.