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\title
Math 262 -- Linear Algebra\vskip 10pt
Spring 2000 (Exam 1)
\endtitle


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\vskip 20pt

1) Define $\det A$ and state the basic properties of the function 
$\det: M_{n\times n}(F)\to F$.

\vskip30pt

2) Let $N\in M_{n \times n}(F)$ be similar to an upper triangular 
matrix whose main diagonal consists entirely of zeros. Show that 
$I+N$ is invertible.

\vskip30pt

3) A matrix $A\in M_{n \times n}(F)$ is said to have a \it square 
root \rm if there exists a \noindent matrix $X\in M_{n \times n}(F)$ 
such that $X^2=A$.

\vskip10pt
\noindent a) Show that if $F=\Bbb R$ and $n$ is odd then $A=-I$ does 
not have a square root.
\vskip10pt

\noindent b) Find a square root of $A=-I$ when $F=\Bbb R$ and $n=2$.

\vskip30pt

4) Consider the linear operator $T:\Bbb R^2 \to \Bbb R^2$, 
$T(x,y,z)=(x, x+2y+3z, x+2z)$.
\vskip10pt

\noindent a) Write down the characteristic polynomial of $T$.

\vskip10pt

\noindent b) Find the eigenvalues of $T$.

\vskip10pt

\noindent c) Describe the eigenspace of $T$ correponding to the \it 
double \rm eigenvalue.

\vskip10pt

\noindent d) Is $T$ diagonalizable? Explain.

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