Matrix multiplication has been standardized so that:

• The left matrix is interpreted as a collection of row vectors
• The right matrix is interpreted as a collection of column vectors
• The dot product is taken between each row of the left matrix and each column of the right matrix

Let the left matrix be $A$, and the right matrix be $B$.

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🧩 Structure of AB

• Each row of $AB$ is formed from the rows of $A$
• Each column of $AB$ is formed from the columns of $B$

This reflects how matrix multiplication contracts across shared dimensions—row of $A$ meets column of $B$.

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🔁 Transposing AB

When we transpose $AB$, we swap its rows and columns:

• Originally: → row from $A$, column from $B$

• After transpose: → row from $B$, column from $A$

Axis Role Flip Transposing $AB$ reverses the roles of $A$ and $B$ in how they contribute to the output matrix.

🔄 What Happens to A and B

To reconstruct this transposed structure:

• $A$ and $B$ must swap places to match the new row-column pairing
• Each must also be transposed individually to preserve the original elements while flipping their orientation

This results in:

• Left matrix: $B^\top$ (formerly the right matrix)
• Right matrix: $A^\top$ (formerly the left matrix)

Their product gives the transposed form:

$$ (AB)^\top = B^\top A^\top $$

Semantic Reversal Transpose flips both the order and the orientation of the original matrices to preserve dot product semantics.

✅ Final Statement

$$ \boxed{ \phantom{\big(} (AB)^\top = B^\top A^\top \phantom{\big)} } $$