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

[perihelion]

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.

 

[lanedance]

you reasoning seems sound to me...

 

[VeeEight]

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.