# Linear Algebra - Linear Algebra

46:

The eigen vectors of a real symmetric matrix corresponding to different eigen values are

 A. orthogonal B. singular C. non-singular D. none of these Answer Report Discuss Option: A Explanation : Let A be a real symmetric matrix, therefore AT=A Let α1 and α2 be different eigen values of matrix A, and X1 and X2 be the corresponding vectors, then AX1= α1X1 and AX2 = α2X2 Taking transpose of the second equation  (AX2)T=  (α2X2) X2TAT= α2.X2T 2X2T But AT-A Post multiply by X1, we get XT2AX1 = a2 X2T X1 But AX1 = a1X1   XT2 a1X1 = a2 X2T X1  (a1 - a2) X2TX1 = 0 Since a1  a2, a1 - a2  0  X2TX1 = 0 i.e. X2 and X1 are orthogonal. Click on Discuss to view users comments. Write your comments here:
47:

An arbitary vector X is an eigen vector of the matrix

 A. (0,0) B. (1,1) C. (0,1) D. (1,2) Answer Report Discuss Option: B Explanation : Since matrix is triangular, the eigen values are a, a, b. (X1,X2,X3) is an arbitary eigen vector, say corresponding to 1, then X2 X3 being not zero, we have X1 = X1 ; aX2 =  X2 which gives a = 1 and bX3 = X3 which gives b - 1  (a, b) = (1, 1) Click on Discuss to view users comments. Write your comments here:
48:

For which value of k, the following system is consistent?

2x-5ky+6z=0

kx+2y-2z=0

2x+2y-kz=0

 A. 1 B. 2 C. 3 D. 5 Answer Report Discuss Option: B Explanation : Equations are consistent, if rank of A and that of k are equal. But in this case it is always true. Hence the equations will have a trivial solution if  IAI  0 Therefore only non- trivial solution will exist if  IAI = 0  2(-2k+4) + k(-k2+4+6)(2k-4) = 0   -5k3+ 20k - 4k + 8 + 12k - 24 = 0  5k3 - 28k + 16 = 0  5k3 - 10k2 + 10k2 - 20k - 8k + 16 = 0  (5k2+ 10k - 8)(k - 2) = 0 Click on Discuss to view users comments. Write your comments here:
49:

The value of  λ for which the equations

2x + y + 2z = 0

x + y + 3z = 0

4x + y + λz = 0

have non-zero solution, is

 A. 2 B. 4 C. 6 D. 8 Answer Report Discuss Option: D Explanation : Equivalent matrix equation is  In order that the given system of equations may have non-zero solution, the rank of A should be less than 3. This requires that  Interchanging R1 and R2 By (R2 - 2R1) and (R3 - 4R1), Click on Discuss to view users comments. Write your comments here:
50:

The system of equations

a1x + a2y = 0

b1x + b2y = 0

where a1, a2 , b1, bare real numbers,

has a non-trivial solutions if

 A. a1b1 = a2b2 B. a1b2 = a2b1 C. a1a2 = b1b2 D. none of these Answer Report Discuss Option: B Explanation : Equations are conistent only if,  a1b2 = b1a2 Click on Discuss to view users comments. Write your comments here: