Math 205a Quiz 8 Worksheet - Bates College - 2007
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1Math 205A
Quiz 8, page 1
November 30, 2007
NAME
1. Let A ∈ M
. We say A is a diagonalizable matrix if and only if there exist two matrices P and D
n×n
both in M
such that:
n×n
1A. P is what kind of a matrix?
1B. D is what kind of a matrix?
1C. A equals what product in terms of P and D?
65
6
30
3
, and 5 has
2. If A =
20
3
10
, then A has eigenvalues 3 and 5. An eigenvector for 3 is
1
6
120 12
55
multiplicity 2. Use this information and the Diagonalization Theorem to diagonalize the matrix A. (Just
find P and D.)
P =
D =
3. Give an example of a matrix S ∈ M
which is invertible, but is not diagonalizable, and explain
2×2
why S has these two properties.
a
a
4. Suppose Q ∈ M
and Q has the form
where a is some real number.
2×2
a
a
4A. Find the determinant of Q.
det=
4B. Is Q invertible?
Circle one: Y
N
4C. Find and simplify the characteristic polynomial of Q.
polynomial is
4D. Find the eigenvalues of Q along with their multiplicities. eigvals & multiplicities
4E. Find a basis for the eigenspace of each eigenvalue.
4F. Is Q diagonalizable? Why or why not?
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