Download Math 205A Quiz 08: Diagonalization and Eigenvectors and more Exercises Linear Algebra in PDF only on Docsity!
Math 205A Quiz 08 page 1 November 21, 2008 NAME
- Let A =
; then A is diagonalizable and has only two eigenvalues, 4 and 5. Find
matrices P and D which represent a diagonalization of A. Even though only one of them is used in the diagonalization, find both P −^1 and D−^1.
You may find it useful that
is row equivalent to
, and
is row equivalent to
Show all your work.
P = P −^1 = D= D−^1 =
Math 205A Quiz 08 page 2 November 21, 2008 NAME.
- Find a 2 × 2 matrix A that has v =
[ 2
]
and w =
[ 4
]
as eigenvectors with associated eigenvalues 3 and 1, respectively. Show all your work, including any matrices you create in order to do this problem. CIRCLE your final matrix A.
- Mark each statement TRUE if it is always true, and FALSE if there are counter examples. Let A be an n × n square matrix. a. A−^1 exists ⇒ A is diagonalizable. b. A−^1 exists ⇒ A does not have 0 as an eigenvalue. c. A is diagonalizable ⇒ A−^1 exists. d. If 0 is not an eigenvalue of A then A is not singular.
