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\noindent
{\Large Math 211 \hfill Homework 5 \hfill LaTex file}

\medskip\noindent
{\bf 1.}
If $A,B,C$ are square matrices such that $A$ and $B$ are invertible such that
$ABC = I$.

(a)  Show that $BCA = I$.

(b) Give $2\times 2$ examples showing that $BAC$ may not be identity even if 
$ABC = I_2$.

\medskip
\noindent{\bf 2.}
Compute the product of two partitioned matrices:
$$\left[\matrix{I & 0 \cr -E & I}\right]\left[\matrix{W&X\cr Y&Z}\right].$$

\medskip
\noindent{\bf 3.}
Suppose
$$\left[\matrix{A&B\cr 0&I}\right]\left[\matrix{X&Y&Z\cr 0&0&I}\right]=
\left[\matrix{I&0&0\cr 0&0&I}\right].$$
Find formulas for $X$, $Y$, and $Z$ in terms of $A$
and $B$.



\medskip\noindent
{\bf 4.} If $A$ is $n\times n$ such that $A^2 = I_n$.
Show that $M^2 = I_{2n}$ if  $M = 
\left[\matrix{A & O_n \cr I_n & -A\cr}\right].$

\medskip\noindent
{\bf 5.} Let $A =  
\left[\matrix{A_{11} & O & O \cr O & A_{22} & O \cr O & O & A_{33} \cr}\right]$.

(a) If  
$A_{11}, A_{22}, A_{33}$ are invertible matrices, show that $A^{-1} = 
\left[\matrix{A_{11}^{-1} & O & O \cr O & A_{22}^{-1} & O \cr O & O & A_{33}^{-1} \cr}\right]$.

[Here the $O$'s in the partitioned matrix is the zero matrices of appropriate sizes.]


(b) Compute $A^{-1}$ if $A_{11} = \left[\matrix{1 & 2 \cr 3 & 5\cr}\right]$
$A_{22} = 
\left[\matrix{7 & 8 \cr 5 & 6\cr}\right]$,
$A_{33} = [2]$.

[Here $A^{-1}$ should be a $5\times 5$ matrix.]

\medskip
\noindent{\bf 6.}
We have the following $LU$ factorization of the matrix
$$A = 
\left[\matrix{2 & -6 & 4 \cr -4 & 8 & 0 \cr 0 & -4 &6\cr}\right]
=\left[\matrix{1 & 0& 0 \cr -2& 1 & 0 \cr 0 & 1 & 1\cr}\right]
\left[\matrix{2 & -6 & 4 \cr0&  -4 & 8 \cr 0 & 0 & -2\cr}\right].$$
Solve the equation $A{\bf x} = {\bf b} = \left[\matrix{2 \cr -4 \cr 6\cr}\right] $ by
first solving (a) $L{\bf y} = {\bf b}$, and then solving (b) $U{\bf x} = {\bf y}$. 



\medskip
\noindent{\bf 7.}
Find the LU factorization of 
$$A = \left[\matrix{-5&0&4\cr 10&2&-5\cr 10&10&16}\right],$$

by applying elementary operations to yield $[A | I_n]$ to $[U | L^{-1}]$.

\medskip
\noindent
{\bf 8.} Suppose $B$ is invertible and $A = BC$.
Show that the reduced row echelon form of  $[B \ A]$ is $[I \ C]$.

[Hint: We can find elementary matrices $E_1, \dots, E_k$ such that
$E_k \cdots E_1[B \ A] = [I \ C]$ according 

to the row operations applied
to $[B \ A]$. Then, ...]

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