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1. Let, solve Ax = b using Cramer’s rule and verify
your answer is correct by checking whether Ax = b is satisfied.
2. Let A be a n × n matrix, prove the following three statements are all equivalent:
(a) Ax = 0 has nontrivial solutions (solutions other than 0).
(b) The determinant of A is zero.
(c) 0 is an eigenvalue of A.
3. Let A ∈ Fm×n,m ≥ n with F = R or C be of full rank, prove that the normal equation A∗Ax = A∗b to the least squares problem minkAx − bk2 has a unique solution for any b ∈ Fn .
