- #1

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**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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- Thread starter sara_87
- Start date

- #1

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

find det(B^T)

(B^T is B transpose)

have none!

help would be greatly appreciated

- #2

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- #3

cristo

Staff Emeritus

Science Advisor

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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.

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:

- #4

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it's the same!

so the determinant of det(B^T) =det(B)=-3

so the determinant of det(B^T) =det(B)=-3

- #5

cristo

Staff Emeritus

Science Advisor

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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!!

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