H13-311_V3.5 · Question #257
The matrix produced by the exchange of matrix A rows and columns is called the transpose of A. What are the correct properties of matrix transposition? (Multiple Choice)
The correct answer is A. (A T ) T = A B. (A+ 8) T = AT +BT C. (!EA) T = !EAT. Transposition has three fundamental properties confirmed by A, B, and C: double-transposing any matrix recovers the original (the "involution" rule), transposing a sum distributes across the addends, and scalar multiples factor cleanly outside the transpose. These follow…
Question
The matrix produced by the exchange of matrix A rows and columns is called the transpose of A. What are the correct properties of matrix transposition? (Multiple Choice)
Options
- A(A T ) T = A
- B(A+ 8) T = AT +BT
- C(!EA) T = !EAT
- D(AB} T = A T +BT
How the community answered
(38 responses)- A74% (28)
- D26% (10)
Explanation
Transposition has three fundamental properties confirmed by A, B, and C: double-transposing any matrix recovers the original (the "involution" rule), transposing a sum distributes across the addends, and scalar multiples factor cleanly outside the transpose. These follow directly from the row-column swap definition applied term by term.
Option D is wrong because it gives (AB)^T = A^T + B^T - which uses addition instead of multiplication and ignores order reversal. The actual rule is (AB)^T = B^T A^T (reversed order, still multiplication). This is a classic trap because matrix multiplication is order-sensitive.
Memory tip: Think "BOAT" - Before Other Ain Transpose - when transposing a product, reverse the order of the factors. For everything else (sums, scalars, double transpose), the operation passes through cleanly without any tricks.
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