Update main.tex

Author: annadavismath

MAT-0050/main.tex

@@ -297,20 +297,22 @@ \section*{Practice Problems}
 
 
 
-\begin{problem}\label{prob:oroveinverseofid}
-Prove Properties \ref{item:inverseofid} and \ref{item:inverseofinverse} of Theorem \ref{th:invprop}.
-\end{problem}
+% \begin{problem}\label{prob:oroveinverseofid}
+% Prove Properties \ref{item:inverseofid} and \ref{item:inverseofinverse} of Theorem \ref{th:invprop}.
+% \end{problem}
 
-\begin{problem}\label{prob:proveinverseka}
-Prove Properties \ref{item:inversekA} and \ref{item:inversetranspose} of Theorem \ref{th:invprop}.
-\end{problem}
+% \begin{problem}\label{prob:proveinverseka}
+% Prove Properties \ref{item:inversekA} and \ref{item:inversetranspose} of Theorem \ref{th:invprop}.
+% \end{problem}
 
 \begin{problem}\label{prob:illustratematinverse} Use the following invertible matrices to illustrate Properties \ref{item:inverseofinverse}, \ref{item:inverseofproduct}, \ref{item:inversekA} and \ref{item:inversetranspose} of Theorem \ref{th:invprop}.
 $$A=\begin{bmatrix}2&0\\4&-3\end{bmatrix}\quad\text{and}\quad B=\begin{bmatrix}2&-1\\1&5\end{bmatrix}$$
 \end{problem}
 
 \begin{problem}\label{prob:solvesysbyinverses}
 In Example \ref{ex:inverse3} we found the inverse of $$A=\begin{bmatrix}1&-1&2\\1&1&1\\1&3&-1\end{bmatrix}$$
+to be 
+$$A^{-1}=\begin{bmatrix}2&-5/2&3/2\\-1&3/2&-1/2\\-1&2&-1\end{bmatrix}$$
 
 Use $A^{-1}$ to solve the equation
 $$A\vec{x}=\begin{bmatrix}3\\-2\\4\end{bmatrix}$$
@@ -328,11 +330,11 @@ \section*{Practice Problems}
 \end{problem}
 
 
-\begin{problem}\label{prob:inverseformula}
-Use the row-reduction method to prove Formula \ref{form:detinverse} for a nonsingular matrix.  Show that if $ad-bc=0$ then $A$ does not have an inverse.
-\begin{hint}After going through the row reduction, try it again, considering the possibility that $a=0$.
-\end{hint}
-\end{problem}
+% \begin{problem}\label{prob:inverseformula}
+% Use the row-reduction method to prove Formula \ref{form:detinverse} for a nonsingular matrix.  Show that if $ad-bc=0$ then $A$ does not have an inverse.
+% \begin{hint}After going through the row reduction, try it again, considering the possibility that $a=0$.
+% \end{hint}
+% \end{problem}
 
 
 \begin{problem}\label{prob:useformforinv}
@@ -344,24 +346,23 @@ \section*{Practice Problems}
 
 \end{problem}
 
-\emph{Problems \ref{prob:notinv1}-\ref{prob:notinv3}}
-
-For each matrix below refer to  Formula \ref{form:detinverse} to find the value of $x$ for which the matrix is not invertible. 
-
 \begin{problem}\label{prob:notinv1}
+Find the value of $x$ for which the matrix is not invertible. 
 $$\begin{bmatrix}1&2\\4&x\end{bmatrix}$$
 Answer: $x=\answer{8}$
 \end{problem}
 
-\begin{problem}\label{prob:notinv2}
-$$\begin{bmatrix}3&2\\3&x\end{bmatrix}$$
-Answer: $x=\answer{2}$
-\end{problem}
+% \begin{problem}\label{prob:notinv2}
+% Find the value of $x$ for which the matrix is not invertible. 
+% $$\begin{bmatrix}3&2\\3&x\end{bmatrix}$$
+% Answer: $x=\answer{2}$
+% \end{problem}
 
-\begin{problem}\label{prob:notinv3}
-$$\begin{bmatrix}5&4\\-5&x\end{bmatrix}$$
-Answer: $x=\answer{-4}$
-\end{problem}
+% \begin{problem}\label{prob:notinv3}
+% Find the value of $x$ for which the matrix is not invertible. 
+% $$\begin{bmatrix}5&4\\-5&x\end{bmatrix}$$
+% Answer: $x=\answer{-4}$
+% \end{problem}
 
 \begin{problem}\label{prob:cancelprop}
 Suppose $AB=AC$ and $A$ is an invertible $n\times n$ matrix. Does it
@@ -374,14 +375,14 @@ \section*{Practice Problems}
 \end{problem}
 
 
-\begin{problem}\label{prob:Asquaredid}
-Give an example of a matrix $A$ such that $A^{2}=I$ and yet $A\neq I$
-and $A\neq -I.$
-\end{problem}
+% \begin{problem}\label{prob:Asquaredid}
+% Give an example of a matrix $A$ such that $A^{2}=I$ and yet $A\neq I$
+% and $A\neq -I.$
+% \end{problem}
 
-\begin{problem}\label{prob:invofsymm}
-Suppose $A$ is a symmetric, invertible matrix.  Does it follow that $A^{-1}$ is symmetric?  What if we change ``symmetric" to ``skew symmetric"?  (See Definition \ref{def:symmetricandskewsymmetric}.)
-\end{problem}
+% \begin{problem}\label{prob:invofsymm}
+% Suppose $A$ is a symmetric, invertible matrix.  Does it follow that $A^{-1}$ is symmetric?  What if we change ``symmetric" to ``skew symmetric"?  (See Definition \ref{def:symmetricandskewsymmetric}.)
+% \end{problem}
 
 \begin{problem}\label{prob:sumofinvertible} Suppose $A$ and $B$ are invertible $n\times n$ matrices.  Does it follow that $(A+B)$ is invertible?
 \end{problem}