g_edgar said:
mlarson9000 said:
epenguin said:
The determinant of a matrix represents the scaling factor of the linear transformation described by that matrix. In other words, it tells us how the matrix affects the size of the vectors it operates on. The first determinant, det(A), refers to the original matrix A, while the second determinant, det((C^-1)AC), refers to the transformed matrix, where C^-1 is the inverse of C and A is the original matrix.
This equality is important because it allows us to calculate the determinant of a transformed matrix without having to perform a time-consuming and error-prone calculation. By using the inverse of C, we can simplify the calculation and still get the same result as if we had calculated the determinant directly from the original matrix.
A matrix is invertible if and only if its determinant is non-zero. In other words, if the determinant of a matrix is zero, it is not possible to find an inverse matrix for it. This is because the determinant represents the scaling factor of the linear transformation, and a scaling factor of zero means no transformation is occurring, making it impossible to reverse the transformation.
No, this equality only holds true for square matrices and invertible matrices. If a matrix is not square, it does not have a determinant. And if the matrix is not invertible, we cannot use its inverse to simplify the calculation of the determinant.
This equality is a direct result of the property of determinants that states det(AB) = det(A)det(B). By using the inverse of C, we can rewrite the transformed matrix as C^-1C, which simplifies to the identity matrix, leaving us with det(A) = det(A), which is always true.