MATH348 - March 1, 2010 NAME:
Exam I - 50 Points - 50 minutes SECTION:
In order to receive full credit, SHOW ALL YOUR WORK. Full credit will be given only if all
reasoning and work is provided. When applicable, please enclose your final answers in boxes.
1. (10 Points)
(a) True/False : Mark each statement as either true or false.
i. Suppose that Ax =b, where A∈Rn×n, has no solutions. The corresponding homogeneous system,
Ax =0, has only the trivial, x=0, solution.
ii. If A∈Rm×nhas a row of zeros then Ax =0always has infinitely-many solutions.
iii. It is impossible for a vector to be in both the null-space and column-space of a matrix.
iv. If the dimension of the column-space of An×nis nthen Ax =0has only the trivial solution.
v. The system Ax =0, where A∈R3×4has only the trivial solution.
(b) Short Response : Provide a short justification of your conclusion.
i. Suppose Vn×nis a matrix whose columns form a basis for Rn. What can be said about the
determinant of V?
ii. Suppose that λ= 0 is an eigenvalue of A. What can be said about A−1?
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