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One more

  • Thread starter sara_87
  • Start date
763
0
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
 
What is the relationship between the 2 determinates we are looking at here? Does 'transposing' the matrix effect the determinate? if so, how?
 

cristo

Staff Emeritus
Science Advisor
8,056
72
Well, to derive it, consider a general 3x3 matrix [tex] \left(\begin{array}{ccc}
a&b&c\\d&e&f\\g&h&i \end{array}\right) [/tex] and expand the determinant

[tex]
\left|\begin{array}{ccc}
a&b&c\\d&e&f\\g&h&i \end{array}\right|=
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}
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:
763
0
it's the same!
so the determinant of det(B^T) =det(B)=-3
 

cristo

Staff Emeritus
Science Advisor
8,056
72
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!!
 
763
0
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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