I wont to learn the logic behind a determinant

In summary, the conversation discusses the logic behind a determinant and its relevance in solving matrix equations. The determinant is defined as the ratio of volume change and can be derived from the exterior algebra. It is important in determining if a matrix has an inverse and can be used to solve equations with non-trivial solutions. The conversation also includes a link to a specific problem involving the determinant.
  • #1
paul-martin
27
0
I won't to learn the logic behind a determinant, the math isn’t so hard you do that then you do that, you don’t need to think.

But if I gone solve dynamic problems, then I must understand how a determinate work. Why do I get the information I want when I take the determinant?

Do anyone got an link which can help me in this?

Kindly paul-Martin...
 
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  • #2
I'm not sure that there is a "how" for the determinant's working. The determinant is defined to be the ratio of volume change of the unit n-cube. Do you mean why does it have the formula it does? Well, you could probably show that it were true geometrically, but that isn't very illuminating. If you know anything about exterior algebras, it can be derived from the fact that the n'th degree component of an n dimensional vector space's exterior algebra is 1 dimensional and the determinant is the induced image of the linear transformation on this component, but I don't know how useful that is to you. It is multiplicative and can be shown to be (essentially) unique and so on.

I don't see what this has to do with dynamics though.

Perhaps you could explain a little more what level you're at and what you mean by "how it works"
 
  • #3
What do YOU mean by "the information I want"?
 
  • #4
Well the problem i gott are something like this. (i have by accident wrote b instead of w)

http://img47.exs.cx/img47/4681/Determinant.jpg

Thx for any help given Paul-M.
 
  • #5
I will confess I was bemused for a moment by "angel speed"! (Calculating how fast an angel can fly? Is that before or after dancing on the head of a pin? :smile: ) I think you meant "angular speed" but anyway, you are seeking a value of w so that the matrix equation Ax= 0 has a non-trivial solution (A depending on w).

The "logic" of the situation is this: If A has an inverse, then we could solve the equation Ax= 0 for the unique solution x= A-10= 0. That is, if A has an inverse, then the equation has only the trivial solution. In order to have a non-trivial solution, A must not have an inverse.
A matrix has an inverse if and only if its determinant is not 0 and so does not have an inverse if and only if its determinant is 0. The equation Ax= 0 has a non-trivial solution if and only if det(A)= 0. You correctly set det(A)= 0 and correctly solved for w.
 

FAQ: I wont to learn the logic behind a determinant

What is a determinant?

A determinant is a mathematical value that can be calculated for a square matrix. It is used to represent certain properties of the matrix, such as its invertibility and the solutions to linear equations represented by the matrix.

Why is the determinant important in linear algebra?

The determinant is important in linear algebra because it helps us understand the properties of matrices, such as whether they are invertible or singular, and what the solutions to linear equations represented by the matrix could be. It also plays a crucial role in calculating eigenvalues and eigenvectors.

How do you calculate the determinant of a matrix?

The determinant of a matrix can be calculated by using various methods, such as the cofactor expansion method, the Gaussian elimination method, or the LU decomposition method. Each method involves a series of mathematical operations on the matrix elements to arrive at the final determinant value.

What is the significance of the sign of the determinant?

The sign of the determinant tells us about the orientation of the vectors in the matrix. If the determinant is positive, it means that the vectors are oriented in a counterclockwise direction, while a negative determinant indicates a clockwise orientation. This information is important in applications such as calculating the area of a polygon or determining if a transformation preserves orientation.

Can you explain the geometric interpretation of a determinant?

The determinant of a matrix can be thought of as the scaling factor of the transformation represented by the matrix. It tells us how much the area or volume of a shape changes when it is transformed by the matrix. A determinant of 1 indicates no change in area or volume, while a determinant of 0 means that the transformation collapses the shape into a lower-dimensional space.

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