Math 1b Practical — Problem Set 5Due 3:00pm, Monday, February 14, 20111. (20 points) Problems 3.10, 3.11, and 3.12 on page 86 of LADW.2. (15 points) Show that 4 is an eigenvalue of the following matrix. (Do not do anynontrivial calculations, but do explain briefly.)⎛⎜⎜⎜⎜⎜⎝5111 0 71511−2 1001151 π −31115 6 71010 1 0823423−3/4⎞⎟⎟⎟⎟⎟⎠3. (10 points) What are the signs of the permutations 4753126 and 7453126?4. (15 points) An n × n matrix A can be stored as substantially fewer than n2numbers,if it has small rank and we are willing to do some computations. For example, if we knowthat A has rank 1, then we can reconstruct A from a nonzero row a and a list of n scalarssuch that the corresponding multiples of r are the rows of A in that order.In general, prove that if A has rank r,thenA = BC where B is an n × r matrix andC is an r × n matrix.(So we may reconstruct A from the 2rn entries of B and C, and when e.g. r =3and n = 100, this represents a substantial reduction in required memory. Sometimes a100 × 100 matrix can be well “approximated” by a rank 3 matrix or some other small rankmatrix, and we will talk about this later.)5. (20 points) LetM =⎛⎜⎝001210110310x1x2x3x4⎞⎟⎠.(i) Expand the determinant of M by the last row, and write det(M) as a “linear form” (x1,...,x4)=ax1+ bx2+ cx3+ dx4in the xi’s. Check that (a) = 0 for each of the topthree rows a of M.(ii) In general, let be the linear form in variables x1,...,xngiven as (x1,...,xn)=det⎛⎜⎝Ax1x2··· xn−1xn⎞⎟⎠where A is an (n − 1) × n matrix of scalars. Explain why is not identically zero (i.e. notall coefficients are zero) if and only if the rank of A is n − 1.6. (20 points) LetA =⎛⎜⎜⎜⎜⎜⎝733333373333337333333733333373333337⎞⎟⎟⎟⎟⎟⎠.(i) What is the characteristic polynomial of A? (Use row operations, by hand.)(ii) For each eigenvalue λ of A, find a basis of the corresponding
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