[SOLVED] MATH307 Individual Homework23

Read textbook pages 135 to 142, pages 126 to 128 before working on the homework problems. Show all steps to get full credits.

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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:
   • Ax = 0 has nontrivial solutions (solutions other than 0).
   • The determinant of A is zero.
   • 0 is an eigenvalue of A.
3. Let A ∈ F^m×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 .