# Determinant of Block Matrix

• Sep 17th 2011, 07:47 AM
H12504106
Determinant of Block Matrix
Let $\displaystyle M=\begin{pmatrix} A & B \\ O & C\end{pmatrix}$, 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.)
• Sep 18th 2011, 04:01 AM
zoek
Re: Determinant of Block Matrix
Quote:

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

$\displaystyle det(M_1) \neq det(M)$.

Quote:

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 $\displaystyle n$ - using cofactor expansion - where $\displaystyle n$ is the type of $\displaystyle A$ i.e. $\displaystyle A$ is an $\displaystyle n\times n$ matrix.