If A is a matrix of order \(m \times n\) and B is a matrix such that \(AB^T\) and \(B^TA\) are both well-defined matrices, then order of matrix B is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(m \times n\)
A product PQ is defined only when the number of columns of P equals the number of rows of Q. Let B be of order \(r \times s\), so \(B^T\) is of order \(s \times r\).
\(AB^T\): \((m \times n)(s \times r)\) is defined, so \(n = s\)
\(B^TA\): \((s \times r)(m \times n)\) is defined, so \(r = m\)
So B is of order \(m \times n\), the same order as A.
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