E231 HW 1, due Friday, September 16, 5:00 PMFor this assignment, you may use the fact that if A and B are square matrices of the samedimension, then det(AB) = det(A) det(B). We will prove this on Monday, September12, using some ideas from the row echelon form results.1. Prove Fact 13 on the Matrix Algebra handout.2. Prove Fact 15 on the Matrix Algebra handout.3. Show, using induction, that if k 6= 0, then for all integers n > 0"cos θ k sin θ−1ksin θ cos θ#n="cos (nθ) k sin (nθ)−1ksin (nθ) cos (nθ)#4. Suppose A is n × n, C is m × n and D is m × m. Show, by induction on n, thatdet"A 0n×mC D#= det(A) det(D)5. If A is n × n, and AT= −A, show thatdet(A) = −1ndet(A)What does this imply when n is odd? Give a simple example.6. Suppose B is n × m C is m × n and D is m × m. Fill in the ? to make the equalityvalid. Be sure to verify that the dimensions are correct."InBC D#="? 0n×mC ?#"? ?0 D − CB#7. If A is n × m and B is m × n show thatdet (In+ AB) = det (Im+ BA)Hint: Start with matrices like"InA−B Im#8. Suppose A is n × n, B is n × m C is m × n and D is m × m. Assume that A isinvertible. Show thatdet"A BC D#= det(A) det(D − CA−1B)19. Find det(M ) whereM = I10+123...910h1 2 3 · · · 9 10i10. Consider A and B as in problem 7. Show that (In+ AB) is invertible if and onlyif (Im+ BA) is invertible (easy). In this case, also showB (In+ AB)−1= (Im+ BA)−1B11. Suppose A and B are n × n matrices. Show that if A, B and A + B are invertible,then A−1+ B−1is invertible, and find an expression for (A−1+ B−1)−1involvingA, B, A + B and their inverses.12. Show that"1 −1 11 1 2#does not have a left inverse.13. Consider the linear equation Ax = b, where A is n × m, b is n × 1 and x is m × 1.Show that(a) If A is left-invertible, then there is at most one solution x to Ax = b. Givean example of a left-invertible A (and some b) for which there is no solution(b) If A is right-invertible, then there is a solution x to Ax = b (although theremay be many). Give an example of a right-invertible A (and some b) forwhich there are many
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