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Department of Aerospace Engineering AE602 Mathematics for Aerospace Engineers Assignment No. 3 3.1 From π΄π΅ = πΆ find a formula for π΄β1 . Do the same from ππ΄ = πΏπ. 3.2 If the inverse of π΄2 is π΅, show that the inverse of π΄ is π΄π΅. (Thus π΄ is invertible whenever π΄2 is invertible.) 3.3 Use the Gauss-Jordan method to invert 1 π΄1 = [1 0 0 0 2 β1 0 0 0 1 1], π΄2 = [β1 2 β1], π΄3 = [0 1 0 1 0 β1 2 1 1 1 1] 1 3.4 Find the inverses (in any legal way) of 1 0 0 0 1 π΄1 = 0 0 2 0 0 3 0 0 ο , π΄2 = 1 2 1 0 2 ο 3 0 0 [4 0 0 0] [ 0 0 0 0 0 1 ο 0 , π΄3 = 3 1 4 ] a b 0 0 c d 0 0 0 0 a b [0 0 c d] . 3.5 Show that for any square matrix π΅, π΄ = π΅ + π΅ π is always symmetric and πΎ = π΅ β π΅ π is always skew-symmetricβwhich means that πΎ π = βπΎ. Find these matrices when π΅ = 1 3 [ ], and write π΅ as the sum of a symmetric matrix and a skew-symmetric matrix. 1 1 3.6 Under what conditions on its entries is π΄ invertible, if π π΄ = [π π π π 0 π 0] 0 or π π΄ = [π 0 π π 0 0 0] ? π 3.7 Compute the symmetric πΏπ·πΏπ factorization of 1 π΄ = [3 5 3 12 18 5 18] 30 And π΄=[ π π π ] π 3.8 Find the inverse of 1 1 1 4 π΄= 1 3 1 [2 0 0 0 1 0 0 1 0 1 3 1 2 1 1 ] 2 3.9 Which descriptions are correct? The solutions π₯ of π₯1 1 1 1 π₯ 0 π΄π₯ = [ ] [ 2] = [ ] 1 0 2 π₯ 0 3 form a plane, line, point, subspace, nullspace of π΄, column space of π΄. 3.10 Compute an πΏπ factorization for 1 2 0 1 π΄ = [0 1 1 0 ] . 1 2 0 1 Determine a set of basic variables and a set of free variables, and find the general solution to π΄π₯ = 0. What is the rank of π΄? 3.11 For the matrix π΄=[ 0 1 4 0 0 2 8 0 ], Determine the echelon from π, the basic variables, the free variables and the general solution to π΄π₯ = 0. Then apply elimination to π΄π₯ = π, with components π1 and π2 on the right side; find the conditions for π΄π₯ = π to be consistent (that is, to have a solution) and the general solution. What is the rank of π΄? 3.12 Carry out the same steps, with π1 , π2, , π3 , π4 on the right side, for the transposed matrix 0 0 π΄= 1 2 4 8 [0 0] 3.13 (a) Find all solutions to 2 1 2 3 4 x1 x2 ππ₯ = [ 0 0 1 2 ] x3 0 0 0 0 [ x4 ] 0 = [0 ] 0 (b) If the right side is changed from (0,0,0) to (π, π, 0), what are the solutions? 3