Thread: Determinant of Block Matrix

Let , where A and C are square matrix. Show that det(M)=det(A)det(C)

I have the following idea but i dont have a concrete proof. We can perform elementary row operations on the matrix M. Eventually A and C will be a upper triangular matrix. Hence det(A) and det(C) will just be the product of their diagonal entries. On the other hand, M will also become an upper triangular matrix after the elementary row operations. Therefore, det(M)=product of its diagonal entries=product of diagonal entries of A * product of diagonal entries of C=det(A)det(C).

How do i write a concrete proof using this idea or is there any other method of doing it (e.g. cofactor expansion, etc.)

Originally Posted by H12504106

Let , where A and C are square matrix. Show that det(M)=det(A)det(C)

I have the following idea but i dont have a concrete proof. We can perform elementary row operations on the matrix M. Eventually A and C will be a upper triangular matrix. Hence det(A) and det(C) will just be the product of their diagonal entries. On the other hand, M will also become an upper triangular matrix after the elementary row operations. Therefore, det(M)=product of its diagonal entries=product of diagonal entries of A * product of diagonal entries of C=det(A)det(C).

You can perform elementary row operations on the matrix but then you will take a matrix with

.

Originally Posted by H12504106

How do i write a concrete proof using this idea or is there any other method of doing it (e.g. cofactor expansion, etc.)

One method is by induction on - using cofactor expansion - where is the type of i.e. is an matrix.