If A is a matrix of order m × n and B is a matrix such that $AB^T$ and $B^TA$ are both well-defined matrices, then order of matrix B is

m × n

The correct answer is Option (3) → m × n

Let $B$ be of order $r \times s$

• $B^T$ is $s \times r$
• $AB^T$ defined ⇒ $A$ ($m \times n$) × $B^T$ ($s \times r$) ⇒ requires $n = s$
• $B^T A$ defined ⇒ $B^T$ ($s \times r$) × $A$ ($m \times n$) ⇒ requires $r = m$

So: $n = s$ and $r = m$ ⟹ $B$ is of order $m \times n$