Possible Values of Determinant for Idempotent Matrix

In summary, if A is an idempotent matrix, the possible values of its determinant are 0 or 1. This can be determined by using the equation det(A) = det(A)*det(A) and considering two cases: if det(A) is not 0, then it must equal 1, otherwise it can also equal 0.
  • #1
bcjochim07
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Homework Statement


If A is an idempotent matrix (A^2 = A), find all possible values of det(A).


Homework Equations





The Attempt at a Solution


I'm not sure if this is the proper way to show it, but here's what I did:

Since A = A^2, det(A)=det(A^2)
So det(A) = det(A)*det(A)

Considering two cases:
If det(A) is not 0, then
det(A) = 1.

The only other way to satisfy det(A) = det(A)*det(A)
is if det(A) = 0.

So det(A) is either 1 or 0.
 
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  • #2
Sure. If det(A)=x then you have x^2=x so x^2-x=0 so x*(x-1)=0. And, yes, that means x=0 or x=1.
 

1. What is a matrix determinant?

A matrix determinant is a mathematical value that can be calculated for a square matrix. It represents the scaling factor of the matrix and is used to solve systems of linear equations, find areas and volumes, and determine invertibility of matrices.

2. How do you calculate a matrix determinant?

To calculate a matrix determinant, you can use the Laplace expansion method, which involves selecting a row or column of the matrix and multiplying each element by its corresponding minor. These products are then summed together to get the determinant value. Alternatively, you can use Gaussian elimination or other methods depending on the size and type of the matrix.

3. What is the significance of a zero determinant?

A zero determinant means that the matrix is singular, meaning it does not have an inverse. This can occur when the rows or columns of the matrix are linearly dependent, making it impossible to solve certain equations or perform certain operations.

4. Can a matrix determinant be negative?

Yes, a matrix determinant can be negative. The sign of the determinant depends on the number of row or column swaps that are performed during the calculation. If an odd number of swaps are made, the determinant will be negative, and if an even number of swaps are made, the determinant will be positive.

5. How is a matrix determinant used in real-world applications?

Matrix determinants have many practical applications, including in engineering, physics, and economics. They can be used to solve systems of linear equations, model physical systems, and analyze data. For example, in economics, determinants are used to study the impact of changes in supply and demand on market equilibrium.

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