Question

Let A be a 2 x 2 matrix such that ATA equals: 21 For each item below determine if it can be calculated from the information provided. If so, answer the associated question and fill in the associated boxes. Otherwise, place an X in all associated boxes. 1. The minimum value of I Av I subject to the condition Ivi - 1. What is the square of this length? 3 2. The slope of the line which contains the shortest vectors in the image of the unit circle under A. What is this slope? -1 12 For Blank 2 3. A vector in the unit circle such that I Av I is maximized and the y-coordinate is positive. What is the sign of the x-coordinate of v. i.e. is the x-coordinate of v positive or negative? Answer either + or- (or X) in the box provided. + 4. The length 1 Aez 1. What is this length? 5. The rank of A What is the rank? 2

Answer 1

Salg Let A be a 282 matrine 1 2X2 7 2. 1 AT 4. 2 4 J2X2 2. ATA 7 2 2 4 24 2X2 1x 7+ 2 X 2 7x2 + 2xy 2x74 4X2 2x2 4 4X4 4944 14 + 8 14 + 8 4+16 53 22 ATA 20 2X2

(5) Rank of Matrin is equal to the ander of of the highest ander non - ringular sub makrise glven matrin non-singular reden matrix (Alto) of ander n then Rank (A) sn. 2 7 2 ALE 4 2. 4 20 to Given → TAL = (7)(4) – (2) (2) 28-4 Matrix A is non singular of order Rank of A A = 27 ) 7 2 4 P (Standard basis 282 2x1 7 2. Ae 1700 + 2x) 12x ot 4X1 2 fil 2X1 Il neoll - J - AG, AS - brace (ne. (re.) & Ac, Al? trace CHA u 8 trace ( G toare (4+16) 20

<AC, AG (2 value of 20 we get es Whing in on then 20 (28215 . 2 85 - 2. (5) % Ae 215 (3) A vector v in the circle nit. Auf is and y coordinate is positive The sign of H-co-ordinates of v valt y