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\centerline{Mathematics 244 -- Linear Algebra}
\centerline{Additional Review True-False Questions -- Exam 3}
\centerline{April 23, 2004}
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\noindent
For the true statements, give a short proof.  For the 
false statements, give a reason or a counterexample.
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\item{A)}  If $A$ is a $3\times 3$ matrix whose characteristic
polynomial has a double root, then $A$ is not diagonalizable.
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\item{B)}  If $T: {\bf R}^2 \to {\bf R}^2$ and there 
is a non-zero vector $x$ that satisfies
$T^2(x) = 4x$, then $T(x) = \pm 2x$.
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\item{C)}  Every $3 \times 3$ matrix has at least one
eigenvalue $\lambda \in {\bf R}$.
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\item{D)}  If $\lambda$ is an eigenvalue of $(A-sI)^{-1}$, where
$s \in {\bf R}$, then $s + {1\over \lambda}$ is an eigenvalue
of $A$.
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\item{E)}  The eigenvalues of $\pmatrix{a&b\cr b&a\cr}$
are $\lambda = a+b$ and $\lambda = a-b$.
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\item{F)}  If the linear mapping $T : {\bf R}^3 \to {\bf R}^3$
has eigenvalues $\lambda = 1,2$, and $x_1,x_2$ are linearly
independent eigenvectors for $\lambda = 1$, while 
$y$ is an eigenvector for $\lambda = 2$, then $E = \{x_1,x_2,y\}$
is linearly independent.
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\item{G)} If $A$ is an $n\times n$ matrix with linearly
independent rows, then $\det(A) = 0$.  
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\item{H)}  Let $V$ be a vector space of dimension $n$
and $T: V \to V$ be linear.  If $[T]_\alpha^\alpha = I_n$
(the $n\times n$ identity matrix)
with respect to {\it some} basis $\alpha$ for $V$, then 
$T$ is the identity mapping.
\bye   
