A determinant is a value associated with a square matrix that can be used to determine various properties of the matrix, such as whether it is invertible or singular.
Group Theory is a branch of mathematics that deals with the study of symmetry and the properties of groups, which are mathematical structures that describe symmetries.
In Group Theory, there is a concept called the group homomorphism which states that for two groups G and H, if there exists a homomorphism from G to H, then the order of the elements in G is preserved in H. This concept is applied to the matrices A and B, which are elements of the general linear group GL(n). By proving that there exists a group homomorphism from GL(n) to the set of non-zero real numbers, we can show that Det(AB) = Det(A)Det(B).
The general linear group GL(n) consists of all invertible n x n matrices with real entries. It is a group under matrix multiplication.
1. Prove that the determinant function is a group homomorphism from the general linear group GL(n) to the set of non-zero real numbers.2. Show that the determinant of the identity matrix is equal to 1.3. Prove that the determinant of a product of matrices is equal to the product of the determinants of each individual matrix.4. Combine these results to show that Det(AB) = Det(A)Det(B).