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Find the value of ${\text{C}}$ where ${{\text{A}}^{\text{T}}}{{\text{B}}^{\text{T}}}{\text{ = }}{{\text{C}}^{\text{T}}}$.
A. \[{\text{AB}}\]
B. ${\text{BA}}$
C. ${\text{BC}}$
D. ${\text{ABC}}$

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Last updated date: 20th Jun 2024
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Answer
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Hint: In this question we have to chose the correct value of ${\text{C}}$ by using the given relation from the given options. For that we are going to solve using transpose properties ${\left( {{{\text{A}}^{\text{T}}}} \right)^{\text{T}}}{\text{ = A}}$, that is the transpose of the transpose of ${\text{A}}$ is ${\text{A}}$ and ${\left( {{\text{AB}}} \right)^{\text{T}}}{\text{ = }}{{\text{B}}^{\text{T}}}{{\text{A}}^{\text{T}}}$, the transpose of a product is the product of the transpose in the reverse order.

Formulas used:
By transpose properties
${{\text{(AB)}}^{\text{T}}}{\text{ = }}{{\text{B}}^{\text{T}}}{{\text{A}}^{\text{T}}}$
${\left( {{{\text{A}}^{\text{T}}}} \right)^{\text{T}}}{\text{ = A}}$

Complete step by step answer:
It is given that transpose relation in the question, ${{\text{A}}^{\text{T}}}{{\text{B}}^{\text{T}}}{\text{ = }}{{\text{C}}^{\text{T}}}$
We have to find the value of ${\text{C}}$ by solving the given transpose relation.
We are going to solve the given transpose relation by using the transpose properties.
Let us consider the transpose relation,
The transpose of a product is the product of the transpose in the reverse order
${{\text{(BA)}}^{\text{T}}}{\text{ = }}{{\text{A}}^{\text{T}}}{{\text{B}}^{\text{T}}}$
By using the transpose relation into given transpose relation ${{\text{A}}^{\text{T}}}{{\text{B}}^{\text{T}}}{\text{ = }}{{\text{C}}^{\text{T}}}$ we get,
We can rewrite the term ${{\text{A}}^{\text{T}}}{{\text{B}}^{\text{T}}}$ as ${{\text{(BA)}}^{\text{T}}}$,
$ \Rightarrow {{\text{(BA)}}^{\text{T}}}{\text{ = }}{{\text{C}}^{\text{T}}}$
Taking transpose of both sides,
$ \Rightarrow {{\text{((BA}}{{\text{)}}^{\text{T}}}{\text{)}}^{\text{T}}}{\text{ = (}}{{\text{C}}^{\text{T}}}{)^{\text{T}}}$
By the transpose relation that is transpose of a transpose gives the terms
That is, ${\left( {{{\text{A}}^{\text{T}}}} \right)^{\text{T}}}{\text{ = A}}$
That is the transpose of ${{\text{(BA)}}^{\text{T}}}$ is ${\text{BA}}$ (the operation of taking the transpose is an involution) and also for the transpose of ${{\text{C}}^{\text{T}}}$ is ${\text{C}}$. Hence we get, ${\text{BA = C}}$

$\therefore $ The option B (${\text{BA}}$) is the correct answer.

Note:
The problem is a simple problem. Here we just use the transpose properties only. It gives the required solution to the question. Although we have to mind the properties of the transpose. We have used transpose properties. The operation of taking the transpose is an involution. The transpose respects addition. The determinant of a square matrix is the same as the determinant of its transpose.