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MATH 307 FALL 2002 1. Let 1 A= 1 EXTRA CREDIT 1 2 1 3 Find all matrices B for which AB is the 2 × 2 identity matrix. Is A nonsingular? Explain. 2. Assuming that v3 6= 0 in the matrix 0 A = v3 −v2 v2 −v1 0 −v3 0 v1 find the kernel of A. 3. What can the rank of a 2 × 3 matrix be? Give an example of a matrix having each possible rank. 4. Let V be the two dimensional subspace of R4 spanned by [4, 4, 4, 1]T , [1, 1, 1, 1]T and let P be the orthogonal projection onto V . Find an orthonormal basis of V and then obtain the matrix of P . 5. Let 6 4 ~b = 0 −3 2 2 1 4 A= 0 1 2 −2 Solve Ax = ~b in the least square sense. 6. Let 1 A= 3 1 1 2 0 4 2 2 Using Cramer’s rule, calculate the (1, 3) entry of A−1 as a quotient of two determinants. (Recall that the the third column of A−1 is the solution of A~x = ~e3 .) 7. Let v be any nonzero n × 1 matrix, and define A = vv T . Show that A is an n × n matrix of rank 1. 8. Let .75 A= .25 .6 .4 Diagonalize A and then use the result to compute A50 . Give an exact answer, not a calculator approximation. (Suggestion: use the formula in Example 7, page 142.)