Title 17.MATRICES.pdf ✅

17. MATRICES-MCQS TYPE (1.) The number of 3 3 non-singular matrices, with four entries as 1 and all other entries as 0 , is [AIEEE-2010] (a.) Less than 4 (b.) 5 (c.) 6 (d.) At least 7 (2.) Let A be a 2 2  matrix with non-zero entries and let 2 A I = , where I is 2 2  identity matrix. Define Tr( A) = sum of diagonal elements of A and A = determinant of matrix A. Statement-1: Tr 0 ( A) = . Statement-2: A =1. [AIEEE-2010] (a.) Statement-1 is true, Statement-2 is true; Statement-2 is acorrect explanation for Statement-1 (b.) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1 (c.) Statement-1 is true, Statement-2 is false (d.) Statement-1 is false, Statement-2 is true (3.) Consider the following relation R on the set of real square matrices of order 3 . ( ) 1 R A B A P BP , − = = ∣ for some invertible matrix P . Statement-1: R is an equivalence relation. Statement-2 : For any two invertible 3 3 matrices M and 1 1 N MN N M ,( ) 1 − − = − . [AIEEE-2011] (a.) Statement- 1 is true, statement- 2 is false (b.) Statement-1 is false, statement-2 is true (c.) Statement- 1 is true, statement-2 is true; statement-2 is a correct explanation for statement-1 (d.) Statement-1 is true, statement-2 is true; statement-2 is not a correct explanation for statement-1 (4.) Statement-1 : Determinant of a skew-symmetric matrix of order 3 is zero. Statement-2 : For any matrix A, det det ( ) ( ) T A A = and det det (− = − A A ) ( ). Where det(B) denotes the determinant of matrix B . Then [AIEEE-2011] (a.) Statement-1 is false andstatement-2 is true (b.) Statement-1 is true and statement-2 is false (c.) Both statements are true (d.) Both statements are false (5.) If  1 is the complex cube root of unity and matrix 0 0 0 H    =     , then 70 H is equal to [AIEEE-2011]
(a.) 2 H (b.) H (c.) 0 (d.) −H (6.) Let 1 0 0 2 1 0 3 2 1 A     =       if 1 u and 2 u are column matrices such that 1 1 0 0 Au     =       and 2 0 1 0 Au     =       , then 1 2 u u + is equal to [AIEEE-2012] (a.) 1 1 1   −         − (b.) 1 1 0   −   −       (c.) 1 1 1     −       − (d.) 1 1 0   −         (7.) If A is an 3 3 non-singular matrix such that AA A A  =  and 1 B A A− =  , then BB equals [JEE (Main)-2014] (a.) 1 B − (b.) ( ) ' 1 B − (c.) I B+ (d.) I (8.) If 1 2 2 2 1 2 2 A a b     = −       is a matrix satisfying the equation 9/ T AA = , where I is 3 3 identity matrix, then the ordered pair (a b, ) is equal to [JEE (Main)-2015] (a.) (2, 1− ) (b.) (−2,1) (c.) (2,1) (d.) (− − 2, 1) (9.) If cos sin sin cos A       − =     , then the matrix 50 A − when 12   = , is equal to [JEE (Main)-2019] (a.) 1 3 2 2 3 1 2 2   −           (b.) 1 3 2 2 3 1 2 2           −   (c.) 1 3 2 2     −   (d.) 3 1 2 2 1 3 2 2     −         (10.) If cos sin cos sin sin cos 2 sin 2 cos t t t t t t t t t t t e e t e t A e e t e t e t e t e e t e t − − − − − − − −     = − − − +     −   , then A is [JEE (Main)-2019] (a.) Invertible only if t = (b.) Invertible for all t R  . (c.) Invertible only if 2 t  = (d.) Not invertible for any t R  .
(11.) Let 2 2 1 1 1 2 b A b b b b     = +       where b  0 . Then [JEE (Main)-2019] (a.) − 3 (b.) 3 (c.) 2 3 (d.) −2 3 (12.) Let 0 2q r A p q r p q r     = −       − . If 3 T AA I = , then p is [JEE (Main)-2019] (a.) 1 3 (b.) 1 6 (c.) 1 5 (d.) 1 2 (13.) Let A and B be two invertble matrices of order 3 3 . If det 8 ( ) T ABA = and ( ) 1 det 8 AB− = , then ( ) 1 det T BA B− is equal to [JEE (Main)-2019] (a.) 1 (b.) 16 (c.) 1 16 (d.) 1 4 (14.) Let 1 0 0 3 1 0 9 3 1 P     =       and Q qij =     be two 3 3 matrices such that 5 Q P l − = 3 . Then 21 31 32 q q q + is equal to [JEE (Main)-2019] (a.) 10 (b.) 135 (c.) 9 (d.) 15 (15.) Let ( ) cos sin , sin cos A R        − =      such that 32 0 1 1 0 A   − =     . Then a value of  is [JEE (Main)-2019] (a.) 32  (b.) 64  (c.) 0 (d.) 16  (16.) Let the numbers 2, , b c be in an A.P. and 2 2 1 1 1 2 4 A b c b c     =       . If det 2,16 ( A)  , then c lies in the interval [JEE (Main)-2019] (a.) 2,3) (b.) ( ) 3/4 2 2 ,4 + (c.) 3/4   3,2 2 +   (d.) 4,6 (17.) If 1 1 1 2 1 3 1 1 1 78 0 1 0 1 0 1 0 1 0 1           n −     =                     , then the inverse of 1 0 1   n     is [JEE (Main)-2019]
(a.) 1 0 13 1       (b.) 1 13 0 1   −     (c.) 1 12 0 1   −     (d.) 1 0 12 1       (18.) The total number of matrices ( ) 0 2 1 2 1 , , , for 2 1 y A x y x y R x y x y     = −         − which 3 3 T A A I = is [JEE (Main)-2019] (a.) 6 (b.) 3 (c.) 4 (d.) 2 (19.) If A is a symmetric matrix and B is a skewsymmetric matrix such that 2 3 5 1 A B   + =     − , then AB is equal to [JEE (Main)-2019] (a.) 4 2 1 4   −     − (b.) 4 2 1 4   −     − − (c.) 4 2 1 4   − −     − (d.) 4 2 1 4   −     (20.) If 5 2 1 0 2 1 3 1 B       =       − is the inverse of a 3 3 matrix A , then the sum of all value of  for which det ( A) + =1 0 , is [JEE (Main)- 2019] (a.) -1 (b.) 2 (c.) 0 (d.) 1 (21.) Let  be a root of the equation 2 x x + + =1 0 and the matrix 2 2 4 1 1 1 1 1 3 1 A         =       , then the matrix 31 A is equal to [JEE (Main)-2020] (a.) 2 A (b.) A (c.) 3 I (d.) 3 A (22.) If 2 2 9 4 A   =     and 1 0 0 1 I   =     , then 1 10A − is equal to [JEE (Main)-2020] (a.) 6I A− (b.) 4I A− (c.) A I −4 (d.) A I −6 (23.) If the matrices 1 1 2 1 3 4 , adj 1 1 3 A B A     = =       − and C A = 3 , then adjB C is equal to [JEE (Main)-2020] (a.) 16 (b.) 2 (c.) 72 (d.) 8