Question on Matrices: find det(B^T)

[sara_87]

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

 

[HalfManHalfAmazing]

What is the relationship between the 2 determinates we are looking at here? Does 'transposing' the matrix effect the determinate? if so, how?

 

[cristo]

Well, to derive it, consider a general 3x3 matrix $$\left(\begin{array}{ccc}<br /> a&b&c\\d&e&f\\g&h&i \end{array}\right)$$ and expand the determinant

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

Then consider the transposed matrix $$\left(\begin{array}{ccc}<br /> a&d&g\\b&e&h\\c&f&i \end{array}\right)$$ and expand this in a similar way. Compare the two results.

 

[sara_87]

it's the same!
so the determinant of det(B^T) =det(B)=-3

 

[cristo]

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!

 

[sara_87]

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!