Question

Give answers only to the following questions. The process of calculating is NOT required. (a) Find the value(s) of h for which the following set of vectors {v1 = (1,0,0)7, v2 = (h, 1, -h)T, V3 = (1, 22, 3h+1)7} is linearly independent. (b) For which choice(s) of the constant k is the following matrix invertible? 1 1 1 1 1 2 k 1 4 k2 ) (c) A is an n x n matrix. |A| = x, the rank of A is y and p is one of the eigenvalues of A. If we double every entry of A and get a new matrix B, find (i) the determinant of B. (ii) the rank of B. (iii) one eigenvalue of B.

Answer 1

(a) Solm: Here, ing = (1,090) T, Uzz Chel; +h) Vg = (1, 2h, 3h+1) T . Now, let A h it .Now, Let A = To 2h Lo-hi 3h+11 , Theni lal = 3h+1+2h2 : := (zh +1).(n+1) ... The set of vectors Vi= (1,0,0)T, N2= (hig, h) and 3=(1,2h, 3h+1) are linearly independ eat if IAlto se if h ř - I or.h -1 ie if ne 6,-1) 01-1) 0 (-1,00): 15 Now, = tó iki applying Rs. Ru, Resep 1k2-1) – 3(K-1) = (x-1) (K+1) - 3(K-1). 3(K-1) (K+1-3) = (k-1) (K-2) iz K oif klor K2. required choices of k are O . * a 3 2, Now, II Fore, K1, or, K32,

(c) Given that A is a nan matrix and 1A I = x, b(A) = y and p is one of the eigen values of A and B = 2A } and B=2A. Therefore, is the determinant of B - ie Bl=12 A1=2" (A) = 2hr. no The rank of B = n(B) = (2A) =r(A) . m) One eigen value of B is p.