2 Questions concerning determinants

In summary, the conversation discusses the number of even permutations in a 5x5 matrix, as well as using determinants to find the area of a triangle with given sides. The first question asks for a formula to determine the number of even permutations, while the second question suggests using vector cross products to relate to the determinant equation in finding the area of a triangle.
  • #1
jvent051
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Homework Statement



First Question. How many even Permutations does a 5x5 matrix have? In other words how many permutations are there that would make it +1 instead of -1.

Second Question. v= (3,2) w= (4,1) use determinants to find the area of a triangle with sides v, w and v+w

Homework Equations





The Attempt at a Solution



First Question. I know you can write them all out, but is there some kind of formula to find how many permutations there are and determine whether the are even or odd? I assume it has something to do with factorials

Second question. I don't know what to do with the matrix when it is not a square, so how do you find a determinant of a triangle? I could use 1/2bh but the problem asks to use determinants specifically
 
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  • #2
first - how many choices is there for the first row? then say you have chosen first row, how many for the 2nd... and so on

2nd - do you know about vector cross products? the length of a vector cross product is the area of the parallelogram made by the 2 vectors... relate this to the determinant equation
 

FAQ: 2 Questions concerning determinants

1. What is a determinant and why is it important?

A determinant is a mathematical concept that is used to describe the properties and behavior of a matrix. It is a numerical value that can be calculated using the elements of a square matrix. Determinants are important because they provide information about the invertibility and singularity of a matrix, and they are used in various applications such as solving systems of linear equations and calculating volumes and areas in geometry.

2. How do you calculate the determinant of a matrix?

The determinant of a matrix can be calculated using various methods, such as the cofactor expansion method, the Gaussian elimination method, or the LU decomposition method. The most common method is the cofactor expansion method, which involves multiplying the elements of a row or column by their corresponding cofactors and then adding these products together. The resulting sum is the determinant of the matrix.

3. Can determinants be negative?

Yes, determinants can be negative. The sign of a determinant depends on the order of the matrix and the arrangement of its elements. For example, a 2x2 matrix with a negative determinant indicates that the matrix is rotated or reflected in a certain way. In general, a negative determinant indicates that the matrix is not invertible.

4. What happens to the determinant when two rows or columns are swapped?

When two rows or columns of a matrix are swapped, the determinant remains the same in magnitude but changes in sign. This is because swapping two rows or columns is equivalent to multiplying the determinant by -1. This property is known as the antisymmetry property of determinants.

5. Can determinants be used to solve systems of nonlinear equations?

No, determinants can only be used to solve systems of linear equations. Nonlinear equations involve variables with powers other than 1, which cannot be represented as a matrix and therefore cannot be solved using determinants. However, determinants can be used to solve systems of linear equations with nonlinear constraints.

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