When is the Determinant of a Square Matrix Equal to Its Negative?

In summary, the question is asking when det(-A) = -det(A) for a square matrix A of size n. The answer is that this is true when n is odd, as shown by multiplying the size n identity matrix by -1 and then by A. This is due to the property that for a scalar c and matrix A, det(cA) = cn det(A).
  • #1
perihelion
3
0

Homework Statement



Suppose A is a square matrix of size n. When is det(-A) = -det(A)?

Homework Equations



N/A

The Attempt at a Solution



My approach to the problem is to simply multiply the size n identity matrix by -1, then multiplied by A. For example: det((-1)*IdentityMatrix[n]*A) = det((-1)*IdentityMatrix[n])*det(A). At this point I could answer the original question by saying this is true when n is odd. But I get the impression I am overlooking some other very obvious answer or condition, and am wondering if anyone can think of a different approach. Thanks.
 
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  • #2
you reasoning seems sound to me...
 
  • #3
Looks good to me too.

Remember that for scalar c and matrix A, det(cA)=cndet(A) where n is the size of A. Plugging in c = -1 gives your answer.
 

Related to When is the Determinant of a Square Matrix Equal to Its Negative?

1. What is a determinant?

A determinant is a mathematical concept used to determine the unique solutions of a system of linear equations. It is represented by a square matrix and is calculated by a specific formula.

2. How is a determinant calculated?

A determinant is calculated by using a specific formula depending on the size of the matrix. For a 2x2 matrix, it is calculated by multiplying the values in the main diagonal and subtracting the product of the values in the other diagonal. For larger matrices, there are more complex formulas that involve expanding the matrix and calculating determinants of smaller matrices.

3. What is the significance of the determinant?

The determinant is significant because it can determine if a system of linear equations has a unique solution, no solution, or infinitely many solutions. It can also be used to find the inverse of a matrix, which is useful in solving systems of equations and performing transformations in linear algebra.

4. Can a determinant be negative?

Yes, a determinant can be negative. The sign of the determinant depends on the values in the matrix and the specific formula used to calculate it. It is possible for the determinant to be positive, negative, or zero.

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

Determinants have various real-life applications, such as in engineering, physics, economics, and computer graphics. They are used to solve systems of equations in these fields, as well as in calculating areas, volumes, and other mathematical quantities. They are also essential in understanding the behavior of linear systems and transformations.

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