Physics 3210 Spring 2008Problem Set 21. For each of the following systems of equations, find a solution if it exists:(a) x + 2y − 3z = −13x − y + 2z = 75x + 3y − 4z = 2(b) 2x + y − 2z = 103x + 2y + 2z = 15x + 4y + 3z = 4(c) x + 2y − 3z = 62x − y + 4z = 24x + 3y − 2z = 14(d) x + 3y + z = 22x + 7y + 4z = 6x + y − 4z = 12. Let f1, f2and f3be e le ments of F [R] (i.e., the space of all real-valuedfunctions defined on R).(a) Given a set {x1, x2, x3} of real numbers, define the 3×3 matrix F (x) =(fi(xj)) where the rows are labeled by i and the columns are labeledby j. Prove that the set {fi} is linearly independent if the rows of thematrix F (x) are linearly independent.(b) Now assume that each fihas first and second derivatives defined onsome interval (a, b) ⊂ R, and let f(j)idenote the jth derivative of fi(where f(0)iis just fi). Define the matrix W (x) = (f(j−1)i(x)) where1 ≤ i, j ≤ 3 and the rows are labeled by i. Pr ove that {fi} is linearlyindependent if the rows of W (x) are independent for some x ∈ (a, b).(The determinant of W (x) is called the Wronskian of the set of func-tions { fi}.)Hint: Prove this by contradiction. Assume that {fi} is linearly de-pendent for all x ∈ (a, b), and show this implies that the rows of W (x)must be linearly dependent for all x ∈ (a, b).Show that each of the following sets of functions is linear ly indepen-dent:(c) f1(x) = −x2+ x + 1, f2(x) = x2+ 2x, f3(x) = x2− 1.(d) f1(x) = exp(−x), f2(x) = x, f3(x) = exp(2x).(e) f1(x) = exp(x), f2(x) = sin x, f3(x) = cos x.3. Using elementary row operations, find the rank of ea ch of the followingmatrices:(a)1 30 −25 −1−2 3(b)5 −1 12 1 −20 −7 124. Repeat the previous problem us ing elementary column operations.15. A matrix of the forma110 0 · · · 00 a220 · · · 0............0 0 0 · · · annis called a diagonal matrix. In other words, A = (aij) is diag onal ifaij= 0 for i 6= j. If A and B a re both square matrices, we may define thecommutator [A, B] of A and B to be the matrix [A, B] = AB − BA. If[A, B] = 0, we say that A and B commute.(a) Show that any diagonal matrices A and B commute. (Explicitly writeout (AB)ij.)(b) Prove that the only n × n matrices which commute with every n × ndiagonal matrix are diagonal matrices.6. What is the inverse of the diagonal matrix shown in Exercise 5?7. Given the matricesA =2 −11 0−3 4B =1 −2 −53 4 0compute the following:(a) AB.(b) BA.(c) AAT.(d) ATA.(e) Verify that (AB)T= BTAT.8. Given a matrix A = (aij) ∈ Mn(F), the sum of the diagonal elements ofA is called the trace of A, and is denoted by tr A. Thustr A =nXi=1aii.(a) Prove that tr(A + B) = tr A + tr B and that tr(αA) = α(tr A) for anyscalar α.(b) Prove that tr(AB) = tr(BA).9. A matrix A = (aij) is said to be upper-triangular if aij= 0 for i > j. Inother words, every entry of A below the main diagonal is zero. Similarly, Ais said to be lower-triangular if aij= 0 for i < j. Prove that the productof upper (lower) triangular matrices is an upper (lower) triangular matrix.10. A square matr ix S is said to be symmetric if ST= S, and a s quare matrixA is said to be skewsymmetric (or antisymmetric) if AT= − A.2(a) Show that S 6= 0 and A 6= 0 are linearly independent in Mn(F).(b) What is the dimension of the space of all n × n sy mmetric matrices?(c) What is the dimension of the space of all n×n antisymmetric matrices?11. Verify the reduced row-echelon form of the matrix given in the exampleon p.70-71 of the notes.12. Find the inverse of a general 2 × 2 matrixM =a bc d.What constraints are there on the entries of the matrix?13. Find the inverse of each of the following matrices:(a)1 0 22 −1 34 1 8(b)1 2 12 5 21 3