Home › Maharashtra HSC (Class 12) › Mathematics › Algebra › The transpose of the product (AB)^T equals
The transpose of the product (AB)^T equals
AA^T B^T
BAB
CB^T A^T
D(BA)^T
Answer & Solution
Correct answer: C. B^T A^T
1. Transpose reverses the order of multiplication.
2. Proof: ((AB)^T)_{ij} = (AB)_{ji} = Σ_k a_{jk} b_{ki} = Σ_k (b^T)_{ik} (a^T)_{kj} = (B^T A^T)_{ij}.
3. Hence (AB)^T = B^T A^T.
_Source: Maharashtra Balbharati Std XII Mathematics Part 1, Ch 2 "Matrices" §2.4_
Related questions
A rule counting sign changes to bound the number of positive real zeros belongs to:A single point where a rational graph is undefined, shown by an open circle, is a hole or A horizontal line that a graph approaches as inputs grow without bound is a horizontal:A vertical line that a graph rushes towards but never crosses is a vertical:A polynomial with real coefficients has 2 plus 3i as a zero, so it must also have as a zerFor a rational zero, numerators come from factors of the constant term and denominators frA polynomial takes a negative value at 3 and a positive value at 4, so between them it musA graph crosses straight through the x-axis at a zero. That zero has multiplicity that is: