Question on Matrices: find det(B^T)

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Homework Help Overview

The discussion revolves around the properties of determinants in relation to matrix transposition, specifically focusing on a 3x3 matrix B with a known determinant value.

Discussion Character

  • Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the relationship between the determinant of a matrix and its transpose, questioning whether transposing affects the determinant. One participant suggests deriving the determinant of a transposed matrix by expanding both the original and transposed matrices.

Discussion Status

Some participants indicate that the determinant of the transposed matrix is the same as that of the original matrix, with specific reference to the value of -3. However, there is no explicit consensus on the reasoning process, as some responses are affirmations rather than explorations of the underlying concepts.

Contextual Notes

The original poster expresses a lack of initial understanding and seeks help, indicating a need for clarification on the topic. There is also a mention of time constraints related to homework completion.

sara_87
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just one last question on matrices, if you don't mind...

Question:

B is a 3*3 matrix det(B)= -3

find det(B^T)

(B^T is B transpose)

My Answer:

have none!

help would be greatly appreciated
 
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What is the relationship between the 2 determinates we are looking at here? Does 'transposing' the matrix effect the determinate? if so, how?
 
Well, to derive it, consider a general 3x3 matrix [tex]\left(\begin{array}{ccc}<br /> a&b&c\\d&e&f\\g&h&i \end{array}\right)[/tex] and expand the determinant

[tex] \left|\begin{array}{ccc}<br /> a&b&c\\d&e&f\\g&h&i \end{array}\right|=<br /> a\left|\begin{array}{cc}e&f\\h&i\end{array}\right| - b\left|\begin{array}{cc}d&f\\g&i\end{array}\right|+c\left|\begin{array}{cc}d&e\\g&h\end{array}\right|=\cdots[/tex]

Then consider the transposed matrix [tex]\left(\begin{array}{ccc}<br /> a&d&g\\b&e&h\\c&f&i \end{array}\right)[/tex] and expand this in a similar way. Compare the two results.
 
Last edited:
it's the same!
so the determinant of det(B^T) =det(B)=-3
 
sara_87 said:
it's the same!
so the determinant of det(B^T) =det(B)=-3

Correct. And in reply to your other thread, happy new year to you too!
 
thanx, my new years resolution is not to leave 200 questions till the last minute! it's nearly 2 am I'm going to finish off these ten questions...and go to sleep!
 

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