We have a block matrix:
$$ \left[\begin{array}{c|c|c} A & 0 & 0 \\ \hline 0 & B & 0 \\ \hline 0 & 0 & C \end{array}\right] $$
Here $A$, $B$ and $C$ are all permutation matrices of varying sizes, raised to a power. For example, all of the block matrices take on forms such as:
$$ \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \\ \end{bmatrix}^a, \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ \end{bmatrix}^b, \text{ or } \begin{bmatrix} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 1\\ 1 & 0 & 0 & 0 & 0\\ \end{bmatrix}^c, $$
We can suppose that we can only obtain one copy of this matrix. In other words, we are not allowed to use this block matrix more than once. Can we assign corresponding scalars to the matrices $(A \to a, B \to b, C \to c)$ and (using matrix multiplication on the block matrix) get a result $a \cdot b \cdot c$?
WHAT I MEAN BY ASSIGNING SCALARS
Suppose that $A$ is only one of the following:
$$ \begin{bmatrix} 0 & 1 \\ 1 & 0 \\ \end{bmatrix} \text{or} \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} $$
...and $B$ is one of the following:
$$ \begin{bmatrix} 0 & 1 \\ 1 & 0 \\ \end{bmatrix} \text{or} \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} $$
We could say that the first $A$ matrix is equivalent to $1$, and the second $A$ matrix is equivalent to 2. Also, the first $B$ equivalent to 3 and the second to 4. Then if we started with the first $A$ and the second $B$, we would be looking to find a result of $1 \cdot 4$. If we started with the second $A$ and the second $B$ we would be trying to obtain a result $2 \cdot 4$.