Exercise 3.3.10

Question

Let $A$ be a square matrix. Show that block triangular matrices 
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\begin{equation*}
    \begin{bmatrix}
    I  &*  \
    \mathbf{0}  &A
    \end{bmatrix}
    \quad
    \begin{bmatrix}
    A  &*  \
    \mathbf{0}  &I
    \end{bmatrix}
    \quad
    \begin{bmatrix}
    I  &\mathbf{0}  \
    *  &A
    \end{bmatrix}  
    \quad
    \begin{bmatrix}
    A  &\mathbf{0}  \
    *  &I
    \end{bmatrix}     
\end{equation*}
%%%%%%%%%%%%%%%%%
all have determinant equal to $\det A$. Here $*$ can be anything.

Jing Huang · Accepted Answer

\begin{proof}
Considering performing row reduction to make A be triangular, the whole matrix will also be triangular and the rest part on the diagonal is just $I$. Thus the determinant of the block matrix equals to $\det A$.
\end{proof}
	extit{(Problem 3.11 and 3.12 are just applications of the conclusion of Problem 3.10. The hint just tells the answer.)}

Exercise 3.3.10

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Exercise 3.3.10

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