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Math Help - Determinant of Block Matrix

  1. #1
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    Determinant of Block Matrix

    Let  M=\begin{pmatrix} A & B \\ O & C\end{pmatrix}, 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.)
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  2. #2
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    Re: Determinant of Block Matrix

    Quote Originally Posted by H12504106 View Post
    Let  M=\begin{pmatrix} A & B \\ O & C\end{pmatrix}, 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 M but then you will take a matrix M_1 with

    det(M_1) \neq det(M).


    Quote Originally Posted by H12504106 View Post
    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 n - using cofactor expansion - where n is the type of A i.e. A is an n\times n matrix.
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