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Math 54 - Spring 1996 - Bergman - Midterm 1

# Math 54 - Spring 1996 - Bergman - Midterm 1 - FRI 10:24 FAX...

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Unformatted text preview: 10/05/2001 FRI 10:24 FAX 6434330 MOFFITT LIBRARY .001 George M. Bergman Spring 1996, Math 54, Lecture 2 13 February, 1996 10 Evans Hall First Midterm 9:40—11:00 AM 1. (3 points each = 24 points) Compute each of the following if it is deﬁned. If it is undeﬁned, say so. (1) A2, where A: 2 4 (ii) [1(3), where p(x)=x2—x—l and B: [(1) i] 6 2 1 (viii) tr 3 5 2. (18 points) Suppose A and B are an matrices, and KER” avector which is both an eigenvector of A, corresponding to the eigenvalue 2L1, and an eigenvector of B, corresponding to the eigenvalue 112. Prove that x is also an eigenvector of AB, corresponding to the eigenvalue 1171.2. 3. (25 points) Consider the system of linear equations 2x1 + 3132 + 4133 : 6 3x1 + 4x2 + SIB = 8 4x1 + 5x2 + 6X3 2 10 (a) (15 points) Give the augmented matrix for the above system of equations, and ﬁnd the reduced row-echelon form of that matrix (b) (10 points) Find all solutions to the above system of equations. (This is most easily done using the result of (a), but the correct answer gotten by any method will be accepted.) 4. (6 points each = 18 points) Deﬁne (using words, formulas, or both) the following concepts. (21) The inverse of an an matrix A (assuming A has an inverse). (b) The linear transformation TA corresponding to an an matrix A. (Your answer should make clear, implicitly or explicitly, the domain and codomain of this transformation.) (0) The standard basis of R4. all 8 1 eigenvector of this matrix corresponding to that eigenvalue. 5. (15 points) Given that the matrix has 3 as an eigenvalue, ﬁnd an ...
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