If A is matrix of order 3 \( \times \) 4 and B is matrix such that \( A^TB \) and \( BA^T \) are both defined, then the order of B is
[Where \( A^T \) denotes transpose of matrix A]
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 3 x 4
\( A^T \) is of order 4 × 3.
\( A^TB \) is defined only if B has 3 rows (the number of columns of \( A^T \)).
\( BA^T \) is defined only if B has 4 columns (the number of rows of \( A^T \)).
So B is of order 3 × 4.
Why the other options are incorrect
Option (2) 3 x 3 — Incorrect. B needs 4 columns, not 3.
Option (3) 4 x 3 — Incorrect. This is the order of \( A^T \), not of B.
Option (4) 4 x 4 — Incorrect. B needs 3 rows, not 4.