Consider the linear equations below:
(1) Which equations are consistent? Explain. (A linear equation is consistent if it has a
solution.)
(2) For each equation that is consistent, find a solution that has the least (Euclidean)
norm.
(3) Among the equations that are not consistent, determine the ones that have a unique
least squares solution? Explain.
(4) For each equation you found in (3), compute the least squares solution.
Suppose A is a 3×3 real matrix, and satisfies
(A2 + A+ I )(A+ 2I ) = 0 (e)
where I is the 3×3 identity matrix.
(1) Show that A is nonsingular.
(2) Express A−1 as a polynomial of A . Is your expression unique?
(3) Let V be the set of all 3×3 real matrices satisfying (e). Is V a vector space?
Explain.
(4) Assume that the eigenvalues of A are distinct, find det(A−1) , the determinant of
A−1 .
Are the following statements True (T) or False (F)? For each of your answers give a
brief explanation. A correct answer with no justification will receive no credits.
(1) If A and B are similar matrices, then det(A) = det(B) .
(2) If two matrices A and B have the same null space, then they must be either
both singular or both nonsingular.
(3) If y ≠ 0 is such that ATy = 0 , then Ax = y has no solution.
Ten students are asked to solve as many puzzles as possible. Let Xi be the number of
puzzles solved by the i-th student, i = 1, 2,?,10 . Suppose that Xi is a Possion
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