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1. If A = - - - l é ë ê ê ê ù û ú ú ú 0 1 2 1 0 3 2 2 is a singular matrix, then l is (A) 0 (B) -2 (C) 2 (D) -1 2. If A and B are square matrices of order 4 4 ´ such that A B = 5 and A B =a× , then a is (A) 5 (B) 25 (C) 625 (D) None of these 3. If A and B are square matrices of the same order such that AB A = and BA A = , then A and B are both (A) Singular (B) Idempotent (C) Involutory (D) None of these 4. The matrix, A = - - - é ë ê ê ê ù û ú ú ú 5 8 0 3 5 0 1 2 1 is (A) Idempotent (B) Involutory (C) Singular (D) None of these 5. Every diagonal element of a skew–symmetric matrix is (A) 1 (B) 0 (C) Purely real (D) None of these 6. The matrix, A = - - é ë ê ê ù û ú ú 1 2 2 2 1 2 i i is (A) Orthogonal (B) Idempotent (C) Unitary (D) None of these 7. Every diagonal elements of a Hermitian matrix is (A) Purely real (B) 0 (C) Purely imaginary (D) 1 8. Every diagonal element of a Skew–Hermitian matrix is (A) Purely real (B) 0 (C) Purely imaginary (D) 1 9. If A is Hermitian, then iA is (A) Symmetric (B) Skew–symmetric (C) Hermitian (D) Skew–Hermitian 10. If A is Skew–Hermitian, then iA is (A) Symmetric (B) Skew–symmetric (C) Hermitian (D) Skew–Hermitian. 11. If A = - - - - - é ë ê ê ê ù û ú ú ú 1 2 2 2 1 2 2 2 1 , then adj. A is equal to (A) A (B) c t (C) 3A t (D) 3A 12. The inverse of the matrix - - é ë ê ù û ú 1 2 3 5 is (A) 5 2 3 1 é ë ê ù û ú (B) 5 3 2 1 é ë ê ù û ú (C) - - - - é ë ê ù û ú 5 2 3 1 (D) None of these CHAPTER Page 525 LINEAR ALGEBRA 9.1 GATE EC BY RK Kanodia www.gatehelp.com 13. Let A = é ë ê ê ê ù û ú ú ú 1 0 0 5 2 0 3 1 2 , then A -1 is equal to (A) 1 4 4 0 0 10 2 0 1 1 2 - - - é ë ê ê ê ù û ú ú ú (B) 1 2 2 0 0 5 1 0 1 1 2 - - - é ë ê ê ê ù û ú ú ú (C) 1 0 0 10 2 0 1 1 2 - - - é ë ê ê ê ù û ú ú ú (D) None of these 14. If the rank of the matrix, A = - é ë ê ê ê ù û ú ú ú 2 1 3 4 7 1 4 5 l is 2, then the value of l is (A) -13 (B) 13 (C) 3 (D) None of these 15. Let A and B be non–singular square matrices of the same order. Consider the following statements. (I) ( ) AB AB T T T = (II) ( ) AB B A - - - = 1 1 1 (III) adj adj adj ( ) ( . )( . ) AB A B = (IV) r =r r( ) ( ) ( ) AB A B (V) AB A B = × Which of the above statements are false ? (A) I, III & IV (B) IV & V (C) I & II (D) All the above 16. The rank of the matrix A = - - - é ë ê ê ê ù û ú ú ú 2 1 1 0 3 2 2 4 3 is (A) 3 (B) 2 (C) 1 (D) None of these 17. The system of equations 3 0 x y z - + = , 15 6 5 0 x y z - + = , l - + = x y z 2 2 0 has a non–zero solution, if l is (A) 6 (B) -6 (C) 2 (D) -2 18. The system of equation x y z - + = 2 0, 2 3 0 x y z - + = , l + - = x y z 0 has the trivial solution as the only solution, if l is (A) l¹- 4 5 (B) l= 4 3 (C) l¹ 2 (D) None of these 19. The system equations x y z + + = 6, x y z + + = 2 3 10, x y z + +l = 2 12 is inconsistent, if l is (A) 3 (B) -3 (C) 0 (D) None of these. 20. The system of equations 5 3 7 4 x y z + + = , 3 26 2 9 x y z + + = ,7 2 10 5 x y z + + = has (A) a unique solution (B) no solution (C) an infinite number of solutions (D) none of these 21. If A is an n–row square matrix of rank (n - 1), then (A) adj A = 0 (B) adj A ¹ 0 (C) adj A = I n (D) None of these 22. The system of equations x y z - + = 4 7 14, 3 8 2 13 x y z + - = , 7 8 26 5 x y z - + = has (A) a unique solution (B) no solution (C) an infinite number of solution (D) none of these 23. The eigen values of A = - é ë ê ù û ú 3 4 9 5 are (A) ± 1 (B) 1, 1 (C) - - 1 1 , (D) None of these 24. The eigen values of A = - - - - é ë ê ê ê ù û ú ú ú 8 6 2 6 7 4 2 4 3 are (A) 0, 3, -15 (B) 0 3 15 , , - - (C) 0 3 15 , , (D) 0 3 15 , , - 25. If the eigen values of a square matrix be 1 2 , - and 3, then the eigen values of the matrix 2A are (A) 1 2 1 3 2 , , - (B) 2 4 6 , , - (C) 1 23 , , - (D) None of these. 26. If A is a non–singular matrix and the eigen values of A are 2 3 3 , , - then the eigen values of A -1 are (A) 2 3 3 , , - (B) 1 2 1 3 1 3 , , - (C) 2 3 3 A A A , , - (D) None of these Page 526 Engineering Mathematics UNIT 9 GATE EC BY RK Kanodia www.gatehelp.com
