(c) Only when AB = BA d) only when AB BA. find the first column of its inverse,

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1 . Find the symmetric and the skew symmetric part of respectively 2. n n n B B, where & B are two matrix above result true (a) lways (b) Never (c) Only when B = B d) only when B B 3. Find the range of K such that the matrix K 6 has an inverse, 5 4. For the matrix find the first column of its inverse, 5. I X 0 Will give us Non trivial solution when (a) 0 (b) 0 (c) I 0 (d) I 0 6. If the matrix singular than (a) Its determinant and Eigen values are zero (b) Only determinant zero but Eigen values non zero (c) Only Eigen values zero but determinant non zero (d) Determinant zero but inverse ext 7. For a matrix (a) Eigen values and Eigen vector are unique (b) Eigen values are unique but vectors are not (c) Eigen vectors are unique but values are not (d) Both are not unique 8. Find the value of 5, if = The product of the matrices i i i i i i i i equal to The Gate Coach ll Rights Reserved 28, Jia Sarai, New Delhi-6, ,

2 i 3 4 i 0 (a) hermitian matrix (c) Skew hermitian (b) unitary matrix (d) orthogonal matrix. If a Skew symmetric matrix of odd order, then the determinant of (a) (b) 0 (c) (d) Real number 2. 3 I 0 given for a matrix find the value of - (a) I + (b) I (c) I + 3 (d) I 2 3. Inverse of if and B are any two matrix such that B = 0 and non singular, then (a) = 0 (b) B = 0 (c) B = (d) B singular 5. If B = 0 and B 0, then necessarily (a) = 0 (b) B = 0 (c) = 0, B = 0 (d) 0, B 0 6. The rank of the matrix diag (a) (b) 2 (c) 3 (d) 4 7. The rank of the matrix (a) 0 (b) (c) 2 (d) 3 8. if a non zero column vector n x, then the rank of the matrix (a) 0 (b) (c) n (d) n T 9. If a non zero row vector ( x n) then the Rank of the matrix (o) (b) (c)n (d) n T The Gate Coach ll Rights Reserved 28, Jia Sarai, New Delhi-6, ,

3 20. The Rank of the matrix 2, find the value of K, (a) 0 (b) (c) 2 (d) 3 K 0 0 K 0 K 2. The rank of the matrix (a) 3 (b) 2 (c) (d) If then find the Eigen values of the matrix (a) 9, 5 (b) 9, 28 (b)7, 3 (d) 7, I 23. The Eigen values of a matrix are, - 2 and 3, then the Eigen values of I are (a) 2,, 6 (b) -, 8, 3 (c) 3, 3, 3 (d) 3, 3, If each element of a 3 x 3 determinant with value P multiplied by 2, then the value of newly formed determinant (a) P (b) 2P (c) 4P (d) 8P 25. The value of the determinant x x 2 x 4 x 3 x 5 x 8 x 7 x 0 x 4 (a) 0 (b) 2 (C) 2 (d) x The Rank of the matrix (a) 0 (b) (c) 2 (d) What the value of a d j If (a) 6 (b) 0 (c) 6 (d) 22 The Gate Coach ll Rights Reserved 28, Jia Sarai, New Delhi-6, ,

4 28. In a third order determinant, each element of the first column consts of sum of two terms, each element of the second column consts of sum of three terms and each element of the third column consts of sum of four terms, then it can be decomposed into n determinants, where n has the value (a) 2 (b) 24 (c) 8 (d) If the system equations x x ky -z = 0 kx y z = 0 x + y z = 0 Has non trivial solution, then the possible values of k are (a), 2 (b), 2 (c) 0, (d), 30. The system of equation x + 2y 3z = 2 (k + 3) z = 3 (2 k + ) y + z = 2 Inconstent for K equal to (a) 3 (b) ½ (c) (d) 2 3. If the matrix b (a) 2 (b) 0 (c) (d) - not invertible then the value of b 32. Lt Lt II a. singular matrix. Determinant not defined B Non singular matrix 2. Determinant always one C. Real symmetric 3. Determinants zero D. Orthogonal matrix 4. Eigen value area always real 5. Eigen values are not defined Match column 33. The rank of the matrix a a a a a a (a) (b) 2 (c) 3 of these The Gate Coach ll Rights Reserved 28, Jia Sarai, New Delhi-6, ,

5 34. The rank of the matrix x y x y x y when the points (x collinear (a) Greater than 3 (b) Less than 3 (c) Greater than 4 of these y ), (x 2, y 2 ) and (x 3, y 3 ) are 35. Rank of the matrix (a) 3 (b) 2 (c) (d) 0 then the value of equal to 36. Rank of the singular matrix of order n can be at most (a) n (b) n + (b) n (d) Let M = then the rank of M equal to 0 (a) 3 (b) 4 (c) 2 (d) 38. The rank of a matrix, whose every element unity, (a) Greater than one (b) equal to one (c) Zero of these 39. Rank of the matrix I n [where I identity matrix] (a) (b) 0 (c) n 40. 3x + 2y + z = 0 X + 4 y + z = 0 2x + y + 4z = 0 (a) Constent (c) No solution 4. x + y + z = x + y 2z = - 2 2x + 4 y + 7z = 7 (a) Constent (c) Only trivial solution ext (b) Only trivial solution exts (d) Determinant of the coefficient matrix zero (b) Inconstent (d) Infinite number of solution exits The Gate Coach ll Rights Reserved 28, Jia Sarai, New Delhi-6, ,

6 42. x + 2y + 2 z = 2x + y + z = - 2 3x + 2y + 2z = 3 Y + z = 0 (a) No solution (c) Infinite (b) Unique solution 43. Value of P for which the system of equations Px + y = x+ 2 y = 3 x + 3 y = 5 are constent (a) (b) 0 (c) 2 (d) 44. For 2 X 2 matrix E ig e n v a lu e E ig e n v e c to r th e m a trix (a) (b) (d) Eigen vectors of the matrix a re (a) (b) (c) The Eigen values of the following matrix a re (a) 3,3 5 i,6 i (b) 6 5 i, 3 i, 3 i The Gate Coach ll Rights Reserved 28, Jia Sarai, New Delhi-6, ,

7 (c) 3, 3, 5 i i i (d) 3, 3 i, 3 i x y, if the Eigen values are 4 and 8, then (a) x = 4, y = 0 (b) x = - 3, y = 9 (c) x = 5, y = 8 (d) x = -4, y = If an identity matrix then, its charactertic root (a) (b) 0 (c) n 49. (a) 4 a condition (a) (c) the Eigen values of the matrix are real and non negative for the a (b) a (d) 2 6 a 2 2 a Let = - and 3 be the Eigen values and 2 Eigen vectors of a real 2 x 2 matrix R. Given that P = V V, then find P R P 2 and be the corresponding 0 The Gate Coach ll Rights Reserved 28, Jia Sarai, New Delhi-6, ,