Title Matrices Varsity Daily MCQ (Set-B)-With Solve.pdf ✅

3 16. A 9 = A n‡j, g ̈vwUa· A Gi chv©q KZ? [A GKwU eM© g ̈vwUa·] [If A9 = A, what is the rank of the matrix A? [A is a square matrix]] 7 3 8 6 DËi: 8 e ̈vL ̈v: A K + 1 = A GLv‡b K n‡jv, chv©q †m‡ÿ‡Î A 8 + 1 = A1  K = 8 17. M =     1 0 3 –1 n‡j, M n‡jv- [M =     1 0 3 –1 then, M is- ] cÖwZmg (Symmetrical) k~b ̈NvwZ (Nilpotent) mgNvwZ (Idempotent) A‡f`NvwZ (Invoultry) DËi: A‡f`NvwZ (Invoultry) e ̈vL ̈v: M 2 =     1 0 3 –1     1 0 3 –1 =     1 0 0 1 = I M 2 = I ZvB A‡f`NvwZ 18. A =     2 1 3 2 6 4 n‡j, A T Gi μg KZ? [A =     2 1 3 2 6 4 then,what is order of AT ?] 2  3 3  3 3  2 2  2 DËi: 3  2 e ̈vL ̈v: A Gi μg 2  3 A T Gi μg 3  2 19. hw` B GKwU eM© g ̈vwUa· nq, Z‡e I –1BI = ? [If B is a square matrix, then I –1BI =?] I B B –1 IB DËi: B e ̈vL ̈v: †Kv‡bv g ̈vwUa‡·i mv‡_ I g ̈vwUa· ̧Y Ki‡j H g ̈wUa‡·i gv‡bi †Kv‡bv cwieZ©b nq bv| 20.       0 a 2 b a 2 c ab2 0 cb2 ac2 bc2 0 = †KvbwU? [       0 a 2 b a 2 c ab2 0 cb2 ac2 bc2 0 = ?] abc 0 2a3 b 3 c 3 1 DËi: 2a3 b 3 c 3 e ̈vL ̈v: a 2 b 2 c 2       0 b c a 0 c a b 0 = a2 b 2 c 2       0 b c 0 – b c a b 0 [c2 = c2 – c3] = a2 b 2 c 2  a     b c – b c = a3 b 2 c 2 (bc + bc) = 2a3 b 3 c 3 21. C =     a – 4 2 0 a + 2 n‡j, a Gi †Kvb gv‡bi Rb ̈ GwU e ̈wZμgx g ̈vwUa· n‡e? [C =     a – 4 2 0 a + 2 = If , it is exceptional for any value of A will the matrix?] – 4, – 2 – 4, 2 4, 2 4, – 2 DËi: 4, – 2 e ̈vL ̈v: |C| = 0      a – 4 2 0 a + 2 = 0  ( a – 4) (a + 2) = 0  a = 4, – 2 22. A =     1 0 1 1 n‡j, A 2024 = ? [A =     1 0 1 1 then, A 2024 = ?]     2024 0 1 1     1 1 2024 0     1 0 2024 1     1 0 1 1 DËi:     1 0 2024 1 e ̈vL ̈v: A =     1 0 1 1 n‡j, A 2 =     1 0 2 1 ; A3 =     1 0 3 1 A n =     1 0 n 1  A 2024 =     1 0 2024 1 23. hw` A =     2 – 4 – 3 1 , Zvn‡j adj(3A2 + 12A) = ? [If, A =     2 – 4 – 3 1 , the, adj(3A2 + 12A) = ?]     51 81 63 72     26 76 42 84     51 72 63 84     51 84 63 72 DËi:     51 84 63 72 e ̈vL ̈v: A =     2 – 4 – 3 1 3A2 =     48 – 36 – 27 39 12A =     24 – 48 – 36 12 3A2 + 12A =     72 – 84 – 63 51 adj(3A2 + 12A) =     51 84 63 72
4 24.       p q r 6 6 6 r + q p + r p + q wbY©vqKwUi gvb KZ? [       p q r 6 6 6 r + q p + r p + q What is the value of the determinant?] 0 2 1 3 DËi: 0 e ̈vL ̈v:       p q r 6 6 6 r + q p + r p + q = 6      p q r 1 1 1 p + q + r p + q + r p + q + r [c3 = c3 + c1] = 6  0 = 0 25. A =     i – i – i 3 i 7 n‡j, A –1 = ? [A =     i – i – i 3 i 7 then, A –1 = ?]     i 0 0 – i     – i 0 0 1 wbY©q‡hvM ̈ bv     1 – 1 i 1 DËi: wbY©q‡hvM ̈ bv e ̈vL ̈v: A =     i – i – i 3 i 7 |A| = i8 + i2 = (– 1)4 – 1 = 1 – 1 = 0 26. A GKwU 2  2 μ‡gi g ̈vwUa· Ges I GKwU 2  2 μ‡gi g ̈vwUa· n‡j, AI5 Gi gvb †KvbwU? [If A is a 2 × 2 matrix and I is a 2 × 2 matrix, what is the value of AI5 ?] 5A 2A A 3A DËi: A e ̈vL ̈v: I =     1 0 0 1 , In =     1 n 0 0 1 n , I5 =     1 5 0 0 1 5  I 5 =     1 0 0 1 †Kv‡bv g ̈vwUa·‡K GKB μ‡gi GKK ev A‡f`K Øviv ̧Y Ki‡j Av‡Mi g ̈vwU·B cvIqv hvq| 27.     2 1 – 3 – 2 GKwU A‡f`NvwZ g ̈vwUa· n‡j, A‡f`NvwZi m~PK KZ? [     2 1 – 3 – 2 If is a non-differentiable matrix, what is the index of the non-differentiable?] 0 1 2 3 DËi: 2 e ̈vL ̈v:     2 1 – 3 – 2     2 1 – 3 – 2 =     2 1 – 3 – 2 2 =     1 0 0 1 28. hw` A =     k 3 – 2 5 Ges A 2 – 7A + 16I = 0 nq, Z‡e k = ? [If A =     k 3 – 2 5 and A 2 – 7A + 16I = 0 then, k = ?] 0 – 3 2 – 2 DËi: 2 e ̈vL ̈v: A =     k 3 – 2 5 A 2 – (Trace of A matrix)A + (Determinant of A matrix)I = 0  A 2 – (k + 5)A + (5k + 6)I = 0 .............. (i) kZ©g‡Z, A 2 – 7A + 16I = 0 (i) bs I (ii) bs Zzjbv Ki‡j, k + 5 = 7 Ges 5k + 6 = 16  k = 2 29.       1 2 3 x sinx + 2x cosx + 3x y siny + 2y cosy + 3y = ? sin(x – y) cos(x – y) xy sin(x – y) cos(x + y) DËi: sin(x – y) e ̈vL ̈v:       1 0 0 x sinx cosx y siny cosy = sinx cosy – cosx siny = sin(x – y)    R2   = R2 – 2R1 R3 = R3 – 3R1 30. A =     3 2 – 1 2 , B =     – 1 2 – 3 0 I A + 2x = 3B, x =? [A =     3 2 – 1 2 , B =     – 1 2 – 3 0 and A + 2x = 3B, x =?]     – 3 2 – 4 1     – 3 2 – 4 – 1     3 – 2 4 – 1 None of these DËi:     – 3 2 – 4 – 1 e ̈vL ̈v: A + 2x = 3B  x = 1 2 (3B – A)  x = 1 2       3     – 1 2 – 3 0 –     3 2 – 1 2  x = 1 2           – 3 6 – 9 0 –     3 2 – 1 2  x = 1 2     – 6 4 – 8 – 2  x =     – 3 2 – 4 – 1 ---