Title 1A. MATRICES (01 - 22).pdf ✅

NISHITH Multimedia India (Pvt.) Ltd., 1 MATRICES NISHITH Multimedia India (Pvt.) Ltd., JEE ADVANCED - VOL - I MATRICES IMPORTANT POINTS a) If A,B are symmetric matrices and commute then 1 A B , 1 AB , 1 1 A B  are also symmetric matrices b) If a square matrix, which is commutative with every square matrix of the same order for multiplication then it is necessarily a scalar matrix. c) There is no square matrix P of order 3, such that DP-PD is equal to unit matrix, where D is a scalar matrix of order 3. d) If product of two non-zero square matrices is a zero matrix, then both of them must be singular matrices. e) If A,B are symmetric matrices of same order and X AB BA   , y AB BA   then XY YX   0 f) Product of two upper triangular matrices of same order, is also an upper triangular matrix. g) If A,B are two idempotent matrices of same order, then A + B is also an idempotent if AB BA   0 h) If A is idempotent and A B I   then B is idempotent and AB BA   0 i) If A,B are two idempotent matrices then A + B is idempotent if AB = BA = 0 j) If , an idempotent matrix is also a skew symmetric then it is a null matrix k)     T T adj A adjA  is a null matrix for all square matrix A l) If  n m A adj   and  ij n m C C   is a cofactor matrix of A then n 1 C A   m) If A be a square matrix of order 3, such that transpose of inverse of A, is it self then adj adjA    1 n) matrix A, such that 2 A A I   2 and for all n n N   2, then  1 n A nA n I    o) If A,B are square matrices of same order and A  0 then for a positive integer n,   1 1 n n A BA A B A    p) If A is an orthogonal matrix and B = AP (P-non- singular matrix) then 1 PB is also orthogonal matrix. LEVEL -V SINGLE ANSWER QUESTIONS 1. Let A,B,C,D be (not necessarily square) real matrices such that AT=BCD, B T=CDA,CT=DAB and DT=ABC for the matrix S=ABCD which of the following is true a) 2 S S  b) 3 S S  c) 4 S S  d) T SS S  2. Let A and B be two square matrices of the same order such that , m AB BA A O   , n B O for some integers m and n, such that greatest common divisor of m, n is 1. Then the least positive integer ‘r’ such that ( )r A B O   is A) m n  B) m n  C) GCD m n,  D) LCM m n,  3. a b A b a         and 2 , m MA A m N   for some matrix M, then which one of the following is correct? A) 2 2 2 2 m m m m a b M b a         B) 2 2 1 0 ( ) 0 1 m M a b         C) 1 0 ( ) 0 1 m m M a b         D) 2 2 1 ( )m a b M a b b a          
MATRICES 2 NISHITH Multimedia India (Pvt.) Ltd., JEE ADVANCED - VOL - I NISHITH Multimedia India (Pvt.) Ltd., 4. Let 3 2 B A A A I     2 3 where I is a unit matrix and A = 1 3 2 2 0 3 1 1 1            then the transpose of matrix B is equal to (A) 8 14 7 21 1 7 14 21 8            (B) 2 21 14 14 1 21 7 7 8            (C) 1 0 0 0 1 0 0 0 1           (D) 3 1 0 1 1 0 3 1 0           5. If 1 2 1 3 A         and if 6 A KA I   205 then (A) K 11 (B) K  22 (C) K  33 (D) K  44 6. Let A is a 3 3  matrix and 3 3    A a ij . If for every column matrix X, if . . T X A X O and 23 a  2009 then 32 a  ...... (A) 2009 (B) -2009 (C) 0 (D) 2008 7. If A,B are two square matrices such that AB = B, BA =A then (A+B)n ( ) n N  is a) A+B b) 2n (A+B) c) 2n–2 (A+B) d) 2n–1 (A+B) 8. If 3 1 2 3 2             is a symmetric matrix then the value of  is A) 5 B) -4 C) 6 D) -6 9. If Adj B A  and, P Q  1 then   1 1 Adj Q B P . .    (A) APQ (B) PAQ (C) B (D) A 10. If A and B are any two 2  2 matrices , then det (A+B) = 0 does not implies (A) det A + det B = 0 (B) det A = 0 or det B = 0 (C) det A =0 and det B = 0 (D) all the above 11. Let P be a non-singular matrix and 2 n I P P P O      then 1 P  is a) n P b) P c) n 1 P  d) I 12. The point of intersection of the planes 2 x y z      2 3 (1 )(1 )     2 2 2 3 ( 1),3 2 1 x y z x y z                (where  is an imaginary cube root of unity, 2 3 1 0, 1        ) is a) (1, 1, 1) b) (1, –1, 1) c) (1, –1, –1) d) (–1, –1, –1) 13. The number of 3 3  matrices A whose entries are either 0 or 1 and for which the system 1 0 0 x A y z                      has exactly two distinct solutions is A) 0 B) 9 2 1 C) 168 D) 2 14. The system of equations x y z 1 x y z 1 x y z 1                   has no solution , if  is (A) Not - 2 (B) 1 (C) -2 (D) Either -2 or 1 15. One of the values of k for which the planes kx y z    4 0, 4 2 0 x ky z    and 2 2 0 x y z    intersect in a straight line A) 0 B) 1 C) 2 D) 3 16. Let 1 2 3 4 A        and 0 , 0 a B a b N b         then a) there exists exactly one B such that AB=BA b) There exists infinitely many B’s such that AB=BA c) there cannot exist any B such that AB=BA d) there exists more than one but finite number of B’s such that AB=BA
NISHITH Multimedia India (Pvt.) Ltd., 3 MATRICES NISHITH Multimedia India (Pvt.) Ltd., JEE ADVANCED - VOL - I 17. If cos sin 0 1 , , sin cos 1 1 A B                     T C ABA  then T n A C A equals to n N   a) 1 1 0   n     b) 1 0 1   n     c) 0 1 1 n        d) 1 0 n 1        18. If 1 0 1 0 , 1 7 0 1 A I                and 2 A A I   8  , then, the value of  is (A) 7 (B) 8 (C) -7 (D) -8 19. If               0 0 0 b a c a c b A and            2 2 2 ac bc c ab b bc a ab ac B , then 2 ( ) A B  (A) A (B) B (C) I (D) 2 2 A B 20. If 0 0 , 0 a b A a c b c              if 1   1 2 T Q A A   and 2   1 2 T Q A A   . Then 1 2 Q .Q is equal to (A) 3 I (B) O3 (C) A (D) 2 A 21. P is an orthogonal matrix and A is a periodic matrix with period 4, T Q PAP  then T 2021 X P Q P  is equal to A) A B) 2 A C) 3 A D) 4 A 22. The matrix 2 2 2 2 2 2 2 2 2 2 2 2 2 2 b a ab a b a b ab a b a b a b                     is A) Idempotent matrix B) nil potent matrix C) Orthogonal matrix D) Unit matrix 23. If i  1 , 1 5 2 a   , 1 5 2 b   then which of the following matrix is idempotent (A) a i i b        (B) b i i a       (C) a i i b       (D) a b b a       24. Let   / 5 and cos sin A sin cos             , then 11 12 13 14 B A A A A     is (A) singular (B) non-singular (C) symmetric (D) Idempotent 25. The inverse of the matrix 1 0 0 A 1 0 1 a is b c            (A)              1 1 0 1 0 0 ac b c a (B)             1 0 0 1 0 0 b c a (C)            1 1 0 1 0 0 ac b a (D)              0 0 1 0 1 1 c a ac b 26. If   cos sin 0 sin cos 0 , 0 0 1 x x F x x x             cos 0 sin ( ) 0 1 0 sin 0 cos y y G y y y           then Adj F x G y ( ( ). ( )) = (A) F (x) G(-y) (B)     1 1 F x G y   (C)     1 1 G y F x   (D) G y F x ( ) ( )  
MATRICES 4 NISHITH Multimedia India (Pvt.) Ltd., JEE ADVANCED - VOL - I NISHITH Multimedia India (Pvt.) Ltd., 27. If            3 1 1 2 3 0 1 2 a A and  1 A              5/ 2 3/ 2 1/ 2 4 3 1/ 2 1/ 2 1/ 2 c then (A) a  2, c 1/ 2 (B) a  1, c  1 (C) a  1, c  1 (D) a  1/ 2, c  1/ 2 28. If   cos sin 0 A , sin cos 0 0 0                     , then which of the following is not true? (A)     T A , A ,      (B)     1 A , A ,       (C) Adj A , A ,             (D)     T A , A ,      29. If AK = O for some positive integral value of K and (I–A)P = I+A+A2 ............+AK–1 then p is ('O' is a null matrix, 'A' is square matrix of order n) a) –1 b) –2 c) –3 d) 1 30. For the equations x y z    2 3 1, 2 3 2, x y z    5 5 9 4 x y z    (A) there is only one solution (B) there exist infinitely many solutions (C) there is no solution (D) the equations are inconsistent. 31. The system of equations 6x 5y z 0 3x y 4z 0 x 2y 3z 0           has (A) only a trivial solution for R (B) exactly one nontrivial solution for some real  (C) infinite number of nontrivial solutions for one value of  (D) none of these. 32. If 0 1 0 A and B 1 1 5 1                , then the value of  for which A2 = B is [IIT-2003] (A) 1 (B) -1 (C) 4 (D) no real values. 33. If P is a 3 3  matrrix such that 2 T P P I   where T P is the transpose of P and I is 3 3  identity matrix then there exists a column matrix 0 0 0 x X y z                       such that (IIT - 2012) A) 0 0 0 PX            B) PX X  C) PX X  2 D) PX X   34. Let P aij      be a 3 3  matrix and Let Q bij      where 2 i j ij ij b a   for 1 , 3.   i j If the determinant of P is 2 then the determinant of the matrix Q is (IIT - 2012) A) 10 2 B) 11 2 C) 12 2 D) 13 2 35. Let  1 be be a cube root of unity and S be the set of all non-singular matrices of the form 2 1 1 1 a b  c             , where each of a b, and c is either  or 2  . Then the number of distinct matrices in the set S is(IIT - 2011) A) 2 B) 6 C) 4 D) 8 MULTI ANSWER QUESTIONS 36. If the adjoint of a 3 3  matrix P is 1 4 4 2 1 7 1 1 3           then the possible values of the determinent of P is (are) (IIT - 2012) A) -2 B) -1 C) 1 D) 2