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Math 205A Winter 2010 Test 2: Matrix Algebra, Exams of Linear Algebra

Mody University of Science and TechnologyLinear Algebra

The instructions and problems for test 2 of math 205a, a college-level matrix algebra course. Students are required to find eigenvalues and eigenvectors, determine if matrices are invertible or diagonalizable, and perform various vector operations. The document also includes problems on finding determinants and the dimensions of subspaces.

Typology: Exams

2012/2013

Uploaded on 02/27/2013

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Math 205A Winter 10
Test 2 (50 points)
Name:
•Check that you have 6 questions on three pages.
•Show all your work to receive full credit for a problem.
1. (6 points) Let C=
30 00
41−52
00 20
10 12
. Find all the eigenvalues of C.
pf3
pf4
pf5

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Math 205A Winter 10

Test 2 (50 points)

Name:

  • Check that you have 6 questions on three pages.
  • Show all your work to receive full credit for a problem.
  1. (6 points) Let C =

. Find all the eigenvalues of^ C.

  1. (6 points) Suppose 0 is an eigenvalue of a 6 × 6 matrix A and ~u is an eigenvector of A corresponding to the eigenvalue 0. (a) Is A invertible? Explain.

(b) Is ~u an eigenvector of A^2? If so, find the corresponding eigenvalue. If not, explain why not.

  1. (10 points) Let W = Span

(a) Find a basis and the dimension of W.

(b) Find a basis and the dimension of W ⊥^ (the orthogonal complement of W ).

(c) Give a geometric description of W and W ⊥.

  1. (8 points) Let A be a 2 × 2 matrix and B be a 2 × 5 matrix, with det A = −4 and det B = 7.

(a) Find det 3B.

(b) Find det A^3. Is A^3 an invertible matrix? Explain.

(c) What is the largest possible rank of B? What is the smallest possible dimension of Nul B? Provide explanations for your answers.