Question

I have attached the questions and the solutions. Please do all of questions 4 and 5

4.a) A matrix A is said to be orthogonalifA". Arv.equivalently,if AA"-A"Asl If A is anorthogonal matrix with integer entries show that every row of A has exactly one nonzero entry which is equal to ± 1. b) If A and B are invertible matrices of the same size show that adj AB adj B(adj A 5 a) Find all unit vectors parallel to the plane x+2y 3z-5 andthe XY-plane. b) Find the point Ponthe plane x'YH whichis closest i the the origin . 1,4+0,6,-4+1-1,-1,1/s,tc R: ANSWERS 1. a) (1+2t,-8t,-1-t,t)tER b) (-2,8,1) 2. a) k=0vk=1 b) k#0^ k#1 c) there is no such k 7 0 101 b)010 001 4 3. a) A- 4. a) Hint: Use the definition of orthogonal matrix. b) Hint: Write adjA in terms of A 5. a b (1,2,3) V5 V5 6. a) Hint: Use Properties of transpose operation. b) 2 17 5 4 010 00 000 8. a) Hint: Use the theorem. b100000001is the basis and dim V,-3 000-100 0-10 9. a) Hint: Use the theorem. The coordinates are 1,2,3. b) tER-1A-7 10. a) False b) False

Answer 1

Alto use e ん )Plan an a o for all then thuii sum 72 aCA) det CAR) CAR 14 dio he esjub' onel Xy.plm2 0

it is an unit vectr V5 lenon Alto we ) VS 28 19 14