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Unformatted text preview: MA 527, Fall 2011, Purdue University Sample questions for exam 1 with solutions This is a collection of problems that are similar to problems that you will find on the actual exam. The actual exam will be in-class and contain a mix of computation, true/false or multiple choice, and proof. The exam will be closed book and notes, no calculator or other computational aids. Chapter 7: Basic linear algebra: Rank, basis, independence, solving systems, determinants, inverse. Chapter 8: Eigenvalues/vectors: Finding eigenvalues/vectors/basis for eigenspace diagonalization/similarity properties Chapter 4: Systems of differential equations: Solution of systems - homogeneous and nonhomogeneous, critical points, types of crit- ical points/stability Sample problems 1. True/False. No explanation necessary. Note that in mathematics, if a statement is sometimes true and sometimes false, then it is false . (a) If A is diagonalizable, then its eigenvalues are distinct. False. E.g. the matrix I is diagonal, hence diagonalizable, but all eigenvalues are 1. (b) If all the eigenvalues of A equal 2, then there is a matrix P so that P- 1 AP = 2 I . False. E.g., the matrix 2 1 0 2 has both eigenvalues 2 but is defective, hence not diagonalizable. (c) If B = C- 1 AC , then B 3 = C- 1 A 3 C . True. Expand B 3 using the definition given, then cancel the terms CC- 1 . (d) If A- 1 exists, then all the eigenvalues of A are non-zero. True. If A- 1 exists, then det( A ) 6 = 0, so det( A- I ) 6 = 0, so 0 is not an eigenvalue....
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- Spring '08
- Advanced Math