Question on Matrices: find det(B^T)

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In summary, the conversation discussed the relationship between determinants and transposing matrices. It was determined that the determinant of a transposed matrix is equal to the determinant of the original matrix, and in this case, both were equal to -3. The conversation also touched on the importance of not procrastinating on school work.
  • #1
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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  • #2
What is the relationship between the 2 determinates we are looking at here? Does 'transposing' the matrix effect the determinate? if so, how?
 
  • #3
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:
  • #4
it's the same!
so the determinant of det(B^T) =det(B)=-3
 
  • #5
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!
 
  • #6
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!
 

1. What is the purpose of finding the determinant of a transposed matrix?

The determinant of a transposed matrix, also known as the cofactor matrix, is used to determine the properties of a matrix. It can help determine if a matrix is invertible, calculate the area or volume of a parallelogram or parallelepiped, and solve systems of linear equations.

2. How do you find the determinant of a transposed matrix?

To find the determinant of a transposed matrix, you first need to transpose the matrix by switching the rows and columns. Then, you can use the same method as finding the determinant of a regular matrix by using the cofactor expansion method or using row operations to simplify the matrix.

3. What is the difference between finding the determinant of a regular matrix and a transposed matrix?

The only difference between finding the determinant of a regular matrix and a transposed matrix is the order in which the calculations are performed. In a regular matrix, the determinant is calculated by multiplying the elements of a row or column by their corresponding cofactors. In a transposed matrix, the elements are multiplied by their corresponding cofactors in the opposite order.

4. Can you find the determinant of a non-square transposed matrix?

No, the determinant can only be calculated for a square matrix. A non-square matrix does not have the same number of rows and columns, so it is not possible to transpose it and perform the necessary calculations to find the determinant.

5. How can finding the determinant of a transposed matrix be useful in real-world applications?

Finding the determinant of a transposed matrix is useful in a variety of real-world applications, such as computer graphics, economics, and engineering. It can help determine the stability of a structure, calculate the flow of electricity in a circuit, and even compress data in computer algorithms.

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