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Unformatted text preview: Solutions to Prelim 1, Math 2940 Spring 2009 Profs and TAs February 26, 2009 Comments: In these solutions we use the notation e i to represent the vector with a 1 in position i and 0 elsewhere. The dimension of the vector should be obvious from context. For problems that ask for an example of a matrix with some particular properties, the solutions below are pretty general. However, to get full credit on the exam you only need to provide one particular matrix, along with an explanation of why that matrix has the desired properties. (1a) If the rank of a 5 by 3 matrix A is 3, what is the rref( A )? What can you say about the solutions of the linear system A x = b ? Solution : Since A is rank 3, there will be 3 leading 1s in rref( A ), and since there are only 3 columns, this means rref ( A ) = 1 1 1 Since in rref( A ) there is a row which is zero, it is possible for the linear system to be inconsistent, and since there are no 0s on the diagonal there will be no free variables. So the possibilities are no solutions or 1 solution to the linear system. (1b) Let A be a nonzero 2 2 matrix. If A 2 = A 3 , does this imply A = I 2 ? If you think this is true, explain why. If not, give a counterexample. (Hint: It might help to think geometrically.) Solution : The assumption A 2 = A 3 does not imply A = I 2 . For example, if A is a projection onto a line, then A 2 = A , so A 2 = A 3 , but A isnt the identity. A specific example is A = 1 ....
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