Final answer:
A. The determinant of the matrix M, expressed as M = AB^{-1}, equals the determinant of matrix A because the determinant of an inverse matrix is the reciprocal of the determinant of the original matrix.
B. Therefore, the determinant of M simplifies to the determinant of A when multiplied by the determinant of the inverse of B.
C. Understanding this relationship has significant implications for matrix decomposition and the study of linear transformations in higher-dimensional spaces.
D. This is valid only if B is invertible, meaning its determinant is not zero.
Step-by-step explanation:
A. When considering the statement that M = AB^{-1} and proving that det(M) = det(A), it is essential to analyze the properties of determinants in regard to matrix multiplication.
The determinant of a matrix product equals the product of the determinants; therefore, the determinant of M can be expressed as
det(M) = det(AB^{-1}) = det(A)det(B^{-1}).
Since B^{-1} is the inverse of a square matrix B, and the determinant of an inverse matrix is the reciprocal of the determinant of the matrix, we have det(B^{-1}) = 1/det(B).
B. As a result, we multiply det(A) by 1/det(B) which simplifies to det(A), proving that det(M) = det(A).
C. Understanding this relationship has significant implications for matrix decomposition and the study of linear transformations in higher-dimensional spaces.
This decomposition of M into a product of A and the inverse of B can provide useful insights for solving systems of linear equations and for performing changes of basis in vector spaces.
D. It's important to mention that B^{-1} exists only if B is invertible, which implies that det(B) ≠ 0 since a matrix is invertible if and only if its determinant is non-zero.