Hemmer said:Is this enough for a proof?
The proof for this relationship is based on the fact that the determinant is a multiplicative function. This means that for two matrices A and B, det(AB) = det(A)det(B). Using this property, we can expand det(A^n) as det(A*A*A*...*A) and then use the multiplicative property to get det(A)^n.
Yes, the relationship holds for any positive integer power of A. This can be seen by using the same logic as the proof for det(A^n), but with a different number of A matrices multiplied together. For example, det(A^2) = det(A*A) = det(A)*det(A) = det(A)^2.
No, this relationship only holds for square matrices. This is because the determinant of a non-square matrix is not defined. In order for the determinant to be defined, the number of rows and columns in the matrix must be equal.
This relationship allows us to simplify calculations involving determinants of powers of a matrix. Instead of having to calculate the determinant of a matrix raised to a power, we can simply raise the determinant to the same power. This can save time and make calculations more efficient.
No, this relationship only applies to the determinant function. It does not hold for other matrix operations, such as addition, subtraction, or inverse. Each of these operations has its own properties and rules that must be followed.