Q.If $(AB)' = B'A'$, where $A$ and $B$ are not square matrices, then number of rows in $A$ is equal to number of columns in $B$ and number of columns in $A$ is equal to number of rows in $B$.

Concept: Matrix Multiplication Compatibility — for $AB$ to be defined, the number of columns in $A$ must equal the number of rows in $B$. The transpose property $(AB)' = B'A'$ holds for any conformable matrices.

Reasoning:

1. Let $A$ be $m \times n$ and $B$ be $p \times q$. For $AB$ to exist, we need $n = p$. Then $AB$ is $m \times q$, so $(AB)'$ is $q \times m$.
2. On the right side, $B'$ is $q \times p$ and $A'$ is $n \times m$. For $B'A'$ to exist, we need $p = n$ (already true) and the product is $q \times m$, matching $(AB)'$. …

The core idea here is simple: the transpose of a product equals the product of the transposes in reverse order — that's a theorem, not a condition. It works for any pair of matrices $A$ and $B$ where $AB$ is defined. The statement in the question tries to turn this theorem into a restriction on the shapes of $A$ and $B$, and that's where it goes wrong.

Let's unpack why.

1. What does $(AB)' = B'A'$ actually require?

   For $AB$ to exist, the number of columns in $A$ must equal the number of rows in $B$. If $A$ is $m \times n$ and $B$ is $n \times p$, then $AB$ is $m \times p$.

   Now, $B'$ is $p \times n$ and $A'$ is $n \times m$, so $B'A'$ is $p \times m$ — which is exactly the transpose of $AB$. The equality $(AB)' = B'A'$ holds for every such pair, no extra conditions needed.

2. What does the question claim?

   It says: "number of rows in $A$ is equal to number of columns in $B$ and number of columns in $A$ is equal to number of rows in $B$".

   Let $A$ be $m \times n$ and $B$ be $p \times q$. The claim is:

   • $m = q$ (rows of $A$ = columns of $B$)
   • $n = p$ (columns of $A$ = rows of $B$)

   But for $AB$ to be defined, we only need $n = p$. The condition $m = q$ is not required for multiplication or for the transpose property.

3. Why is the claim wrong?

   Take a concrete counterexample. Let $A$ be $2 \times 3$ and $B$ be $3 \times 4$. Then $AB$ is $2 \times 4$, and $(AB)'$ is $4 \times 2$.

   $B'$ is $4 \times 3$, $A'$ is $3 \times 2$, so $B'A'$ is $4 \times 2$ — it works perfectly.

   But here, rows of $A$ = 2, columns of $B$ = 4 — they are not equal. So the statement is false. …

Method: Separating a Theorem's Requirements From Extra (Unneeded) Conditions

Use this when a statement turns a general theorem into a restriction on matrix shapes. Identify what the theorem actually requires versus what is being added.

Steps

Step 1: State the theorem and its real precondition.

The reversal law $(AB)'=B'A'$ holds whenever $AB$ is defined. With $A$ of order $m\times n$ and $B$ of order $n\times p$, the only requirement is that columns of $A$ equal rows of $B$ (the shared inner dimension).

Step 2: Translate the claimed conditions into order equations.

"Columns of $A$ = rows of $B$" is $n=n$ — already the compatibility condition, so that part is fine. "Rows of $A$ = columns of $B$" is $m=p$ — an additional demand the theorem never makes. …

Common Mistakes

Mistake 1: Believing the reversal law imposes a symmetry on the shapes.

Why it's wrong: $(AB)'=B'A'$ needs only $AB$ to be defined (columns of $A$ = rows of $B$); it never forces rows of $A$ = columns of $B$. Correct approach: separate "needed for $AB$ to exist" from an invented extra condition.

Mistake 2: Confusing "$(AB)'=B'A'$" with "both $AB$ and $BA$ defined." …