B3A001S Pages: 2

PART C
Answer any two questions.
7. (೩) Using Gauss elimination method, find the solution of the system of equations
x+2y—-—z=3,3x —y+2z = 1, 22 - 2) + 32 = 2 andx-y+z=-1 (7)

(0) Find the values of (८ for which the system of equations + 2 + 2 = 1, ‰ + 21 +
32 ‏بيرح‎ and ८ + 51 + 92 = സി be consistent. For each value of ம obtained,

find the solution of the system. (7)
(c) Prove that the vectors (2,3,0). (1,2,0) and (8,13,0) are linearly dependent in 03,
(6)
2 3 -1 -1
, ⋅ 1 -1 -2 -1
8. (a) Find the rank of the matrix 4 = | 3 1 3 -2 (7)
6 3 0 -7
1 0 -1
(0) Find the eigen values and eigen vectors of the matrix}1 2 1 (7)
2 2 3

(c) Write the canonical form of the quadratic form Q(x, y,z) = 3x? + Sy? + 322 -
22 + 2xz—2yz and hence show that Q(x,y,z) > 0 for all non-zero values of

Xs; 2: (6)
2 0 1
9. (a) Diagonalize the matrix 4 = |0 2 | and hence find 44. (7)
1 0 2
3 -1 1
(0) If 2 is an eigen value ൨-1 5 5 without using its characteristic equation,
1 -1 3
find the other eigen values. Also find the eigen values of 43, 47, 4 1, 54, 4 - 31 and
adj A. (7)
(c) Show that / 72 - 30xy + 1 7)2 = 128 represents an ellipse. Also find the equations
of the major and minor axes of the ellipse in terms of x and y. (6)
RK

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