Match List I with List II
| LIST I | LIST II | ||
| A. | A and B are skew symmetric matrices | I. | \((AB)^T=B^TA^T\) |
| B. | A and B are symmetric matrices and AB is symmetric | II. | AB is a symmetric matrix |
| C. | A and B are skew symmetric matrices and AB = BA | III. | \(AB=BA\) |
| D. | For any \(m\times n\) and \(n\times p\) matrices A and B | IV. | A + B is skew symmetric matrix |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → A-IV, B-III, C-II, D-I
The tools are \(A^T = A\) (symmetric), \(A^T = -A\) (skew symmetric) and the reversal law \((AB)^T = B^TA^T\).
A. A and B are skew symmetric matrices
This matches IV. A + B is skew symmetric matrix.
\((A + B)^T = A^T + B^T = -A - B = -(A + B)\)
B. A and B are symmetric matrices and AB is symmetric
This matches III. \(AB = BA\).
\(AB = (AB)^T = B^TA^T = BA\)
C. A and B are skew symmetric matrices and AB = BA
This matches II. AB is a symmetric matrix.
\((AB)^T = B^TA^T = (-B)(-A) = BA = AB\)
D. For any \(m\times n\) and \(n\times p\) matrices A and B
This matches I. \((AB)^T = B^TA^T\).
The reversal law is the only result in List II that holds for every pair of matrices whose product is defined; such a product need not even be square, so II, III and IV cannot hold in general.