If U is an orthogonal matrix,its determinant is equal to 1 or -1.

In summary, an orthogonal matrix is a square matrix with perpendicular rows and columns, and the determinant of an orthogonal matrix is always either 1 or -1. This is because the determinant represents the scaling factor, and in an orthogonal matrix, there is no change in scaling. Therefore, the determinant cannot be 0. The significance of the determinant being 1 or -1 is that it indicates the matrix is orthonormal, which has many practical applications.
  • #1
mcbonov
3
0
Question:
Prove that is U is an orthogonal matrix, then the determinant of U is equal to 1 or -1.
Hint consider the equation U^t = U^-1 and use the properties of the determinant.


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So far I only found out ,since it is invertible ,its determinant is not zero.
I can't go any further than that...
Please help me.
 
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  • #2
welcome to pf!

hi mcbonov! welcome to pf! :smile:

(try using the X2 icon just above the Reply box :wink:)
mcbonov said:
… use the properties of the determinant.

the determinant of the product is the product of the determinants :wink:
 

FAQ: If U is an orthogonal matrix,its determinant is equal to 1 or -1.

What is an orthogonal matrix?

An orthogonal matrix is a square matrix in which all the rows and columns are orthogonal (perpendicular) to each other. This means that the dot product of any two rows or columns is equal to 0.

How do you determine the determinant of an orthogonal matrix?

The determinant of an orthogonal matrix can be determined by multiplying the diagonal elements of the matrix. Since all the rows and columns are orthogonal, the product will always be either 1 or -1.

3. Why is the determinant of an orthogonal matrix always equal to 1 or -1?

This is because the determinant of a matrix represents the scaling factor of the matrix, and in an orthogonal matrix, the rows and columns are perpendicular to each other, so there is no change in the scaling of the matrix. Therefore, the determinant will always be either 1 or -1.

4. Can an orthogonal matrix have a determinant of 0?

No, an orthogonal matrix can never have a determinant of 0 because it would mean that the scaling factor of the matrix is 0, which is not possible.

5. What is the significance of the determinant being 1 or -1 in an orthogonal matrix?

The determinant being 1 or -1 in an orthogonal matrix indicates that the matrix is orthonormal, which means that the rows and columns are not only perpendicular to each other but also have a length of 1. This makes orthogonal matrices useful in many applications, such as in rotations and reflections.

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