Math 205A Quiz 8, page 1 November 30, 2007 NAME
1. Let A∈Mn×n.WesayAis a diagonalizable matrix if and only if there exist two matrices Pand D
both in Mn×nsuch that:
1A. Pis what kind of a matrix? 1B. Dis what kind of a matrix?
1C. Aequals what product in terms of Pand D?
2. If A=
65 6 −30
−20 3 10
120 12 −55
, then Ahas eigenvalues 3 and 5. An eigenvector for 3 is
−3
1
−6
, and 5 has
multiplicity 2. Use this information and the Diagonalization Theorem to diagonalize the matrix A. (Just
find Pand D.)
P=D=
3. Give an example of a matrix S∈M2×2which is invertible, but is not diagonalizable, and explain
why Shas these two properties.
4. Suppose Q∈M2×2and Qhas the form aa
−a−awhere ais some real number.
4A. Find the determinant of Q. det=
4B. Is Qinvertible? Circle one: Y N
4C. Find and simplify the characteristic polynomial of Q. polynomial is
4D. Find the eigenvalues of Qalong with their multiplicities. eigvals & multiplicities
4E. Find a basis for the eigenspace of each eigenvalue.
4F. Is Qdiagonalizable? Why or why not?