Question

Matrix Methods/Linear Algebra: Please show all work and justify the answer!

4. Let A and B be 4 x 4 matrices. Suppose det A = 4 and det(AB) = 20. (a) (4 points) What is det B? (b) (4 points) Is B invertible? Why or why not? (c) (4 points) What is det(A”)? (d) (4 points) What is det(A-")? 5. (6 points) Let A be an n x n invertible matrix. Use complete sentences to explain why the columns of A" are linearly independent. and u = Is {vi, os} a basis for R?? Why or why 6. (6 points) Let vi = not?

Answer 1

J142) g) det (A) = 4, det (AB) = 20 Now, det (AB) = det (A) X det (B). 20 4X det (B) det (B) = 5 b) since det (B) & 0 So the inverse of B exists & hence B is invertible. c) cet (AT) = det (A) 4 d) det (14-1) = det (A) 4

Page No 05= let AT - B * fact! With &,,...,n refersing to the columns AT=B, the equation niv + Hz Va t .- + Hp n = 0 can be rewritten as Bu=0. * fact: If A is invertible, then AT = B les also invertible, then AT = B is dow- equivalent to the identity matrix. & fact: If R is the row reduced version of AT=B, then R and B have the same hull espace. That Is Ru=0 & Br=0 have is amo solution. from the above facts, we conclude that if LGT is invertible, then At is dow equivalent to I, BInce Columns Of I are lincauly independent, so the columner of AT must be linearly independent, 06-) = [1] -2 L-4 Now, cuit (₂ 62 C [ 1 ] + CZ + 2-2 L-4

Date - / Page No C, - 2 = 0 267-462 = 0 1 -2 Ry - 2 Ri 1 e] -210 o 2 - 4 10 Since V & ve are not linearly independent, So, &u , r 3 is not a basis of R².

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