If A is a 2 × 3 matrix and B is a matrix such that \( \mathrm{A^T\,B} \) and \( \mathrm{B\,A^T} \) are both defined. Then order of B is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 2 × 3
A is 2 × 3, so \( \mathrm{A^T} \) is 3 × 2. A product PQ is defined only when the number of columns of P equals the number of rows of Q. Let B be m × n.
\( \displaystyle \mathrm{A^T\,B}:\ (3\times 2)(m\times n)\ \Rightarrow\ m=2 \)
\( \displaystyle \mathrm{B\,A^T}:\ (m\times n)(3\times 2)\ \Rightarrow\ n=3 \)
So B is of order 2 × 3.
Tip: If both PB and BP are defined for a p × q matrix P, then B must be q × p, the order of \( \mathrm{P^T} \). Here P = \( \mathrm{A^T} \), so B has the order of A.