\begin{problem}{MATH 293}{FALL 1981}{PRELIM 1}{2}{}
\prob[*]{Consider the matrices} $$A =  \left[ \begin{array}{ccc}
2&4&1 \\ 1&1&1 \\ 2&3&1
\end{array} \right], B = \left[ \begin{array}{cccc}
1&2&3&4 \\ 2&3&5&6 \\ 1&3&4&6 \\ 1&4&5&8
\end{array} \right].$$
\probpart{Find det $A$ and det $B$.} \probpart{Find $A^{-1}$ and
$B^{-1}$ If they exist.  If you think that either of the inverses
does not exist, give a reason.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1982}{PRELIM 1}{1}{}
\probb[*]{Write the system of equations
 $\begin{array}{ccccccc}
x_{1} & + & 2x_{2} & + & 3x_{3} & = & 1
\\ 2x_{1} & + & 3x_{2} & +& 4x_{3} & = & -2
\\ 3x_{1} & + & 4x_{2} & + & 6x_{3} & = & 0
\end{array}$
in the form $A\vec{x} = B$.} \probpart{Find det $A$ for $A$ in
part $(a)$ above.} \probpart{Does $A^{-1}$ exist?} \probpart{Solve
the above system of equations for $\vec{x}$ = $ \left(
\begin{array}{c} x_{1} \\ x_{2} \\ x_{3}
\end{array} \right)$
}
\probpart{Let $B$ = $ \left[
\begin{array}{ccc} 1&0&1 \\ 0&1&0 \\ 2&-1&1
\end{array} \right]$. Find $A \cdot B$ (i.e. calculate the product $A B$). }
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1991}{PRELIM 3}{4}{}
\prob[*]{Compute the determinants of the following matrices:}
\probpart{$ \left( \begin{array}{ccc} 2&3&0 \\ 3&5&0
\\ 0&0&1 \end{array}
\right) $}
\probpart{ $ \left( \begin{array}{cccc} 2&3&0&0 \\
3&5&0&0
\\ 0&0&-11&-3 \\ 0&0&4&1 \end{array}
\right) $}
\probpart{ $ \left( \begin{array}{cccc} 2&-5&17&31 \\
0&3&9&14
\\ 0&0&-1&7 \\ 0&0&0&4 \end{array}
\right)^{-1} $}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1991}{PRELIM 3}{6}{}
\prob[*]{True/False} \probpart{All three row operations preserve
the absolute value of the determinant of a square matrix.}
\probpart{A singular $n \times n$ matrix has a zero determinant.}
\probpart{If $A$ is a $n \times n$ matrix  $\det \left( A^{t}
\right) = \det (A)$.} \probpart{If each entry in a square matrix
$A$ is replaced by its reciprocal (inverse), producing a new
matrix $B$, then $ \det (B) = ( \det ($A$) )^{-1}.$} \probpart{If
a matrix $A$ is nonsingular, then $ \det ($A$^{-1} ) =( \det
($A$) )^{-1} $ } \probpart{For a square matrix $A$ and a scalar
$k$, $ \det (k A) = k \det (A) $} \probpart{Let $A$ and $B$ by $n
\times n$ matrices
$$ \det \left( \begin{array}{cc}
  A & O_{n} \\
  O_{n} & B
\end{array} \right) = \det (A) \det (B) $$
where $O_{n}$ is a $n \times n$ matrix with all elements equal to
zero. }
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1992}{PRELIM 3}{1}{}
\prob[*]{Compute} \probpart{$ \det \left[ \begin{array}{ccc} 0&2&0 \\
1&0&3
\\ 5&0&8 \end{array}
\right] $} \probpart{$ \det \left[
\begin{array}{cc} \cos \theta & \sin \theta \\ - \sin \theta &
\cos \theta
\end{array}
\right] $}
\probpart{$ \det \left[ \begin{array}{ccccc} 1&1&7&0&0
\\ 1&1&0&3&3 \\ 5&5&1&8&9 \\ 6&6&1&0&1 \\ 6&6&1&0&1
\end{array}
\right] $}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SUMMER 1992}{PRELIM 7\_21}{2}{}
\prob[*]{Compute the following determinants:} \probpart{ $\left|
\begin{array}{cccc}
  1 & 2 & 1 & 1 \\ 0 & 1 & -1 & -1 \\  2 & 3 & 2 & 3 \\  -3 & -5 & 0 & -1
\end{array} \right|$ }
\probpart{ $\left| \begin{array}{ccc}
  3- \lambda & 0 & 1 \\
  2 & 1- \lambda & -4 \\
  1 & 0 & -1- \lambda
\end{array} \right|$ }
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1992}{PRELIM 3}{1}{}
\prob[*]{Compute the determinant of the matrix} $$ A =\left(
\begin{array}{ccccc}
  b & a & a & a & a \\
  b & b & a & a & a \\
  b & b & b & a & a \\
  b & b & b & b & a \\
  b & b & b & b & b
\end{array}
\right) $$
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1992}{PRELIM 3}{4}{}
\prob[*]{Let $A$ be an $n \times n$ matrix.  Assume that it is
known that the equation $Ax = 0$ has nontrivial solutions if and
only if $\det (A) = 0$

Let $$ A = \left( \begin{array}{ccc}
  3-s & 0 & 1 \\
  2 & 1-s & -4 \\
  1 & 0 & -(1+s)
\end{array} \right)$$ where $s$ is an arbitrary scalar.}
\probpart{Compute $\det (A)$} \probpart{Find those values of $s$
for which the equation $Ax=0$ has nontrivial solutions.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1992}{FINAL}{3}{}
\prob[*]{} \probpart{Let $A$ be an $n \times n$ nonsingular
matrix. Prove that $\det \left( A^{-1} \right) =\frac{1}{\det
(A)}$. Hint: You may use the fact that if $A$ and $B$ are $n
\times n$ matrices, $\det(A B) = \det(A)\det(B)$. } \probpart{An
$n \times n$ matrix $A$ has a nontrivial null space.  Find
$\det(A)$ and explain your answer.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1993}{PRELIM 3}{1}{}
\prob[*]{Given the matrix$$ A = \left( \begin{array}{ccc}
  -2 & 1 & 2 \\
  -2 & 2 & 2 \\
  -9 & 3 & 7
\end{array} \right)$$Find $\det A$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1994}{PRELIM 2}{6}{}
\prob[*]{}\probpart{Compute the determinant of the matrix
$A(\lambda) = \left(
\begin{array}{ccc}
  1-\lambda & 1 & 0 \\
  2 & 2-\lambda & 1 \\
  0 & 1 & 2-\lambda
\end{array} \right)$, writing your result as a function of $\lambda$. }
\probpart{Partially check your result by computing the determinant
of $A(0)$, and compare this value with the value of the function
you found in a) when $\lambda=0$ }
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1994}{PRELIM 2}{3}{}
\prob[*]{Compute $$ \det \left[ \begin{array}{ccc}
  1 & -2 & 3 \\
  -3 & 5 & -8 \\
  2 & 2 & 5
\end{array} \right] $$ by the following two methods:}
\probpart{Use row ops to change the matrix into an upperright
triangular matrix with the same $\det$} \probpart{Use cofactors of
entries in the first row }
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1994}{PRELIM 2}{5}{}
\prob[*]{Let $A$ and $B$ be $N \times N$ matrices.}
\probpart{Complete the following statement: $A$ is singular if and
only if $\det(A) = ...$ } \probpart{Use the result of part (a) to
find the value of $\lambda$ for which the matrix $\left(
\begin{array}{cc}
  \lambda-1 & 3 \\
  2 & \lambda-2
\end{array} \right)$
is singular } \probpart{Complete the following statement: $\det(A
B) =$ ... } \probpart{Use the result of part (c) to show that if
$A$ is invertible, $\det \left( A^{-1} \right) = \frac{1}{\det
A}$. (Hint: $A A^{-1} = I$)}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1994}{PRELIM 2}{6}{}
\prob[*]{Compute$$\det\left( \begin{array}{cccc}
  0 & 0 & -1 & 3 \\
  0 & 1 & 2 & 1 \\
  2 & -2 & 5 & 2 \\
  3 & 3 & 0 & 0
\end{array}\right).$$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1994}{PRELIM 3}{3}{}
\prob[*]{Let $A$ be an $n by n$ matrix.  Which of the following is
equivalent to the statement:  \textbf{$A$ is singular?}}
\probpart{The $\det (A) = n$.} \probpart{$A x = 0$ has a
nontrivial solution. } \probpart{The rows of $A$ are linearly
independent.} \probpart{The rank of $A$ is n.} \probpart{The $\det
A = 0$.} \probpart{$A x = B$ has a unique solution $x$ for each
$B$.} \probpart{$A$ has non-zero nullity.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1994}{FINAL}{3}{}
\prob[*]{Evaluate the determinant $\left|\begin{array}{cccc}
  a & b & b & b \\
  a & a & b & b \\
  a & a & a & b \\
  a & a & a & a
\end{array} \right|$ by first using row reduction to convert it to upper triangular form.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1994}{FINAL}{12}{}
\prob[*]{If $A$ is a 3 by 3 matrix and $\det(A) = 3$, then
$\det\left(\frac{1}{2}A^{-1}\right)$ is:}
\probpart{$\frac{1}{24}$} \probpart{$\frac{2}{3}$}
\probpart{$\frac{1}{6}$} \probpart{$\frac{8}{3}$}
\probpart{$\frac{3}{8}$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1994}{FINAL}{13}{}
\prob[*]{If $A B$ is singular, then} \probpart{$\det(A)$ is zero,}
\probpart{$\det(B)$ is zero,} \probpart{$\det(A)$ is zero and
$\det(B)$ is zero,} \probpart{$\det(A)$ is not zero and $\det(B)$
is not zero,}\probpart{either $\det(A)$ is zero or $\det(B)$ is
zero.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1994}{FINAL}{14}{}
\prob[*]{Given the system $\left[ \begin{array}{cc}
  1 & 2 \\
  3 & 3
\end{array} \right] \left[ \begin{array}{c}
  x \\
  y
\end{array} \right] = \left[ \begin{array}{c}
  1 \\
  3
\end{array} \right]$.  With $p = \det \left|\begin{array}{cc}
  1 & 2 \\
  2 & 3
\end{array} \right|$, $q = \det \left|\begin{array}{cc}
  1 & 2 \\
  3 & 3
\end{array} \right|$, $r = \det \left|\begin{array}{cc}
  1 & 1 \\
  2 & 3
\end{array} \right|$, $s = \det \left|\begin{array}{cc}
  1 & 0 \\
  0 & 1
\end{array} \right|$, by Cramer's Rule, the solution for $y$ is given by}
\probpart{$\frac{s}{p}$} \probpart{$\frac{r}{p}$}
\probpart{$\frac{p}{r}$} \probpart{$\frac{q}{p}$}
\probpart{$\frac{r}{q}$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1995}{PRELIM 2}{5}{}
\prob[*]{Let $A$ be a $6 \times 6$ matrix.} \probpart{Which of the
following 3 terms will appear in $\det A$:
$$a_{13}a_{22}a_{36}a_{45}a_{51}a_{64},
a_{15}a_{21}a_{36}a_{45}a_{52}a_{63},
a_{16}a_{25}a_{34}a_{43}a_{52}a_{61}  ? $$} \probpart{For those
which will appear, what will their signs be?} \probpart{How many
such terms will there be in all?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1995}{PRELIM 2}{6}{}
\prob[*]{} \probpart{Calculate the determinant of the matrix $$ A
= \left[\begin{array}{cccc}
  2 & 0 & 1 & -1 \\
  1 & 2 & -1 & 1 \\
  0 & 1 & 1 & -1 \\
  -2 & -2 & 1 & 0
\end{array}
\right]$$} \probpart{What can you say of the solutions to the
equation $$ A x = 0.$$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1996}{PRELIM 3}{9}{}
\prob[*]{Let $A$ be an $n$ by $n$ matrix.  Which of the following
are equivalent to the statement "the determinant of $A$ is not
zero"? You do \textbf{not} need to show any work. " }
\probpart{The columns of $A$ are linearly independent.}
\probpart{The rank of $A$ is equal to $n$. } \probpart{The null
space of $A$ is empty.} \probpart{$A \vec x = \vec b$ has a
unique solution for each $\vec b$ in $\Re^{n}$.} \probpart{$A$ is
not onto.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1996}{FINAL}{12}{}
\prob[*]{The determinant of the matrix below is: $$
\left(\begin{array}{cccc}
  1 & -3 & 1 & -2 \\
  2 & -5 & -1 & -2 \\
  0 & -4 & 5 & 1 \\
  -3 & 10 & -6 & 8
\end{array} \right)$$}
\probpart{1} \probpart{-1} \probpart{2} \probpart{0}
\probpart{none of above}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1997}{FINAL}{2}{}
\prob[*]{If the $\det A = 2$.  Find the $\det A^{-1}$, $\det
A^{T}$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1997}{FINAL}{6}{}
\prob[*]{The equation of a surface S in $\Re^{3}$ is given as $$
\det \left( \begin{array}{cccc}
  x & y & z & 1 \\
  a_{1} & a_{2} & a_{3} & 1 \\
  b_{1} & b_{2} & b_{3} & 1 \\
  c_{1} & c_{2} & c_{3} & 1
\end{array}\right) = 0 $$ where the $a_{i}, b_{i}, c_{i}$ are constants.}
\probpart{Does the point $ a = \left( \begin{array}{c}
  a_{1} \\
  a_{2} \\
  a_{3}
\end{array} \right)$ lie on S?} \probpart{Do the points $ b = \left( \begin{array}{c}
  b_{1} \\
  b_{2} \\
  b_{3}
\end{array} \right)$ and $c = \left( \begin{array}{c}
  c_{1} \\
  c_{2} \\
  c_{3}
\end{array} \right)$ lie on S?}
\probpart{Find a relationship between the coordinates of a, b, and
c such that if this relationship holds, then the origin lies on
$S$. }
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1998}{PRELIM 2}{5}{}
\prob[*]{Use cofactor expansion to compute the determinant
$$\left[
\begin{array}{cccc}
  1 & -2 & 5 & -2 \\
  0 & 0 & 3 & 0 \\
  2 & -6 & -7 & 5 \\
  0 & 0 & 4 & 4
\end{array}\right].$$ At each step choose a row or column that involves the least amount of
computation.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1998}{PRELIM 2}{5}{}
\prob{True or False?} \probpart{The determinant of an $n \times n$
triangular matrix is the product of the entries on the main
diagonal.} \probpart{The cofactor expansion of an $n \times n$
matrix down a column is the negative of the cofactor expansion
along a row.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1998}{PRELIM 2}{2}{}
\prob[*]{Evaluate the determinant of $B = \left[
\begin{array}{cccc}
  1 & 2 & -3 & 4 \\
  -4 & 2 & 1 & 3 \\
  3 & 0 & 0 & 0 \\
  2 & 0 & -2 & 0
\end{array} \right].$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING ?}{PRELIM 2}{3}{}
\prob[*]{} \probpart{Compute $\det \left[ \begin{array}{ccc}
  4 & -7 & 2 \\
  5 & 2 & 0 \\
  3 & 0 & 0
\end{array}\right]$}
\probpart{If $F(x) = \det \left[ \begin{array}{cccc}
  1 & x & x^{2} & x^{3} \\
  1 & 2 & 2^{2} & 2^{3} \\
  1 & 3 & 3^{2} & 3^{3} \\
  1 & 1 & 1 & 1
\end{array} \right]$,
then show that $F(737)$ is not zero. (Hint: How many roots can the
equation $F(x) = 0$ have?) }
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{UNKNOWN}{PRACTICE}{2b}{}
\prob{A $n \times n$ matrix $C$ is said to be orthogonal if $C^t =
C^{-1}. $ Show that either $\det{C} = 1$ or $\det{C} = -1.$ Hint:
$ C C^T = I .$}
\end{problem}
