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

1 Daily-01 [A (Solve Sheet)] wm‡jevm: g ̈vwUa· I wbY©vqK c~Y©gvb: 30 †b‡MwUf gvK©: 0.25 mgq: 20 wgwbU 1. wb‡Pi †KvbwU wecÖwZmg g ̈vwUa·? [Which of the following is asymmetric matrix?]     b 0 0 – b     b – b 0 0     0 b – b 0     0 0 – b b DËi:     0 b – b 0 e ̈vL ̈v: wecÖwZmg g ̈vwUa· nIqvi kZ©: A T = – A  A T =     0 b – b 0 T =     0 – b b 0 = –     0 b – b 0 = – A 2. A I B g ̈vwUa‡·i gvÎv h_vμ‡g 4  3 I 3  2 n‡j, (B + A T ) Gi gvÎv KZ? [The dimension of the A and B matrix is respectively 4  3 and 3  2, then what is the dimension of (B + AT )?] 3  4 4  4 3  3 Indeterminate DËi: Indeterminate e ̈vL ̈v: A T Gi gvÎv 3  4 B Gi gvÎv 3  2  ( B + AT ) Gi gvÎv wbY©q Kiv m¤¢e bq 3.       1 6 3 2 2 1 3 0 4 Gi (1,2) Zg Abyivwki gvb KZ? [       1 6 3 2 2 1 3 0 4 What is the value of (1,2)th term?] 12 10 6 24 DËi: 24 e ̈vL ̈v: (1, 2) Zg fzw3 Gi Abyivwk =    6 3 0 4 = 24 4. A =       2 5 – 4 0 b 8 – 1 2 5 g ̈vwUa‡· Trace Gi gvb 1 n‡j, b = ? [A =       2 5 – 4 0 b 8 – 1 2 5 If its Trace value is 1 in the matrix, b=?] 1 2 – 6 5 DËi: – 6 e ̈vL ̈v: 2 + b + 5 = 1  b = – 6 5.    K + 1  2 12 – 8 g ̈vwUa·wU e ̈wZμgx n‡j, K Gi gvb- [    K + 1  2 12 – 8 If the matrix is exceptional, the value of K-] – 4 – 8 5 2 DËi: – 4 e ̈vL ̈v:     K + 1 2 12 – 8 = – 8K – 8 – 24 = 0  – 8K = 32  K = – 4 6. A GKwU 3 μ‡gig ̈vwUa· Ges |A| = – 8 n‡j, |(2A)–1 | Gi gvb n‡e- [A will be the 3 3 matrix of a sequence and |A| = – 8, |(2A)–1 | its value- ] 1 64 – 1 8 2 1 56 1 – 24 DËi: – 1 8 2 e ̈vL ̈v: |A| = – 8 |(2A)–1 |= 1 2 3  – 8 = – 1 8 2 7.       a 0 0 0 b 0 0 0 c GKwU KY© g ̈vwUa· n‡j, †Kvb k‡Z© † ̄‹jvi n‡e? [       a 0 0 0 b 0 0 0 c If a diagonal matrix, under what condition will it be a scalar?] a = b = c a  b = c a = b + c a + b = c DËi: a = b = c e ̈vL ̈v: eM© g ̈vwUa‡· i  j Ae ̄’v‡b fzw3 ̧‡jv 0 Ges i = j Ae ̄’v‡b fzw3 mgvb n‡j, † ̄‹jvi| 8. hw` GKwU eM© g ̈vwUa· A Ggb nq †h, A 2 – 3AI2 + I2 = 0 Z‡e, A –1 = ? [If a square matrix A is such that A2 – 3AI2 + I2 = 0 then A–1 = ?] A – 3I A 2 + 3I 3I – A 3I – A 2 DËi: 3I – A e ̈vL ̈v: A 2 – 3AI2 + I2 = 0  A 2 – 3AI + I = 0  A – 3I + A–1 = 0 [A–1 Øviv ̧Y K‡i]  A –1 = 3I – A
2 9.       5 1 3 6 2 6 7 3 9 = ? 0 1 2 3 DËi: 0 e ̈vL ̈v:       5 1 3 6 2 6 7 3 9 = 3      5 1 1 6 2 2 7 3 3 = 3  0 [`ywU Kjvg GKB] = 0 10.       3 4 1 – 1 3 0 5 – 2 6 wbY©vq‡K 0 Gi Abyivwk KZ? [       3 4 1 – 1 3 0 5 – 2 6 What is the minor of 0 to determinants?] 26 20 10 – 26 DËi: – 26 e ̈vL ̈v: Ò0Ó Gi Rb ̈ Abyivwk     3 4 5 – 2 = – 26 11.       1   2   2 1  2 1  Gi gvb KZ? [       1   2   2 1  2 1  What is its value?] 1  2 0  DËi: 0 e ̈vL ̈v:       1   2   2 1  2 1  =       1 +  +  2 1 +  2 +   2 + 1 +    2 1  2 1  [c1 = c1+ c2 + c2] =       0 0 0   2 1   1  = 0 12. hw` A =     loge x – loge x – 1 2 Ges det(A) = 2 n‡j, x = ? [If A =     loge x – loge x – 1 2 and det(A)=2, then, x=?] 2 e 2 e log2 DËi: e 2 e ̈vL ̈v: 2loge x – loge x = 2  loge x = 2  x = e2 loge x = b x = eb 13. A = [aij] GKwU 2  2 μ‡gi g ̈vwUa·, †hLv‡b aij = 1 hLb, i  j Ges aij = 0 hLb i = j Zvn‡j, A 2 = ? [A = [aij] is a 2  2 matrix of order, where aij = 1 when, i  j and aij = 0 when i = j then, A2 = ?]     1 1 0 0     1 0 1 0     1 0 1 1     1 0 0 1 DËi:     1 0 0 1 e ̈vL ̈v: kZ©g‡Z, a11 = 0, a12 = 1, a21 = 1, a22 = 0 A =     0 1 1 0 , A2 =     0 1 1 0     0 1 1 0 =     1 0 0 1 14. A =     1 3 2 4 n‡j, A –1 = ? [A =     1 3 2 4 then, A –1 = ?]     4 – 3 – 2 1 1 2     4 – 3 2 1 – 1 2     4 – 3 – 2 1 1 2     4 3 2 1 DËi: – 1 2     4 – 3 – 2 1 e ̈vL ̈v: A –1 = 1 4 – 6     4 – 3 – 2 1 [Using shortcut] = – 1 2     4 – 3 – 2 1 15. A‡f`NvwZ g ̈vwUa‡·i R‡b ̈ wb‡Pi †KvbwU mwVK? [Which of the following is correct for non-differentiable matrix?] A 2 = A A T = – A – A = A A 2 = I2 DËi: A 2 = I2 16. wb‡Pi †KvbwU j¤^ g ̈vwUa·? [Which of the following is orthogonal matrix?]     1 3 – 2 4     5 – 1 3 – 2     2 – 3 3 2     3 5 2 6 DËi:     2 – 3 3 2 e ̈vL ̈v: j¤^ g ̈vwUa· †Pbvi Dcvq: g~L ̈ K‡Y©i fyw3 ̧‡jv GKB n‡j,, †MŠY KY© GKB wKš‘ wPý Avjv`v Ges †MŠY K‡Y©i fzw3 ̧‡jv GKB n‡j,, g~L ̈ KY© GKB wKš‘ wPý Avjv`v| 17. A =     1 0 1 1 n‡j, A 5 = ? [A =     1 0 1 1 then, A 5 = ?]     1 0 0 1     1 0 5 1     0 0 5 1     0 0 0 0 DËi:     1 0 5 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 5 =     1 0 5 1 18. x Gi †Kvb gv‡bi Rb ̈       2 2 1 3 3 2 4 x 3 wbY©vq‡Ki gvb k~b ̈ n‡e? [For which value of x,       2 2 1 3 3 2 4 x 3 Will the value of the determinant be zero?] 4 1 0 3 DËi: 4 e ̈vL ̈v: x Gi cwie‡Z© 4 emv‡j `ywU mvwi GKB n‡e| ZLb wbY©vq‡Ki gvb k~b ̈ n‡e|
3 19.       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 = which one?] 2a2 b 3 c 3 2a3 b 3 c 2 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 20. 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 Indeterminate     1 – 1 i 1 DËi: Indeterminate e ̈vL ̈v: A =     i – i – i 3 i 7 |A| = i 8 + i2 = (– 1)4 – 1 = 1 – 1 = 0 21.       p q r 4 4 4 r + q p + r p + q wbY©vqKwUi gvb KZ? [       p q r 4 4 4 r + q p + r p + q what is the value of determinants?] 0 2 1 3 DËi: 0 e ̈vL ̈v:       p q r 4 4 4 r + q p + r p + q = 4      p q r 1 1 1 p + q + r p + q + r p + q + r [c3 = c3 + c1] = 4  0 = 0 22. hw` A =       2 0 0 0 2 0 0 0 2 nq, Z‡e A 4 Gi gvb KZ? [If A =       2 0 0 0 2 0 0 0 2 , then what is the value of A 4 ?]       2 0 0 0 2 0 0 0 2       8 0 0 0 8 0 0 0 8       32 0 0 0 32 0 0 0 32       16 0 0 0 16 0 0 0 16 DËi:       16 0 0 0 16 0 0 0 16 e ̈vL ̈v: A 4 =       2 4 0 0 0 2 4 0 0 0 2 4 =       16 0 0 0 16 0 0 0 16 KY© g ̈vwUa‡·i †ÿ‡Î, A n =       a n 0 0 0 b n 0 0 0 c n Shortcut: A =       a 0 0 0 a 0 0 0 a n‡j, A n =       a n 0 0 0 a n 0 0 0 a n 23.       1 0 0 0 1 0 0 0 1 g ̈vwUa·wU‡K wb‡Pi †Kvb g ̈vwUa· ejv hv‡e bv? [       1 0 0 0 1 0 0 0 1 Which of the following matrix cannot be called a matrix?] † ̄‹jvi g ̈vwUa· (scalar matrix) A‡f`K g ̈vwUa· (Invariant matrix) KY© g ̈vwUa· (Diagonal matrix) k~b ̈ g ̈vwUa· (null matrix) DËi: k~b ̈ g ̈vwUa· (null matrix) e ̈vL ̈v: KY© g ̈vwUa· (Diagonal Matrix): †h eM© g ̈vwUa‡·i cÖavb K‡Y©i fzw3 ̧wj k~b ̈ ev Ak~b ̈ Ges Ab ̈vb ̈ fzw3mg~n k~b ̈ (A_v©r aij = 0; i  j) Zv‡K KY© g ̈vwUa· e‡j| †hgb: D1 =     5 0 0 2 , D2 =       1 0 0 0 2 0 0 0 3 D3 =       5 0 0 0 0 0 0 0 3 , D4 =       0 0 0 0 0 0 0 0 0 cÖ‡Z ̈KwU KY© g ̈vwUa·| † ̄‹jvi g ̈vwUa· (Scalar Matrix): †h KY© g ̈vwUa‡·i cÖavb K‡Y©i fzw3 ̧wj mgvb Zv‡K † ̄‹jvi g ̈vwUa· e‡j| †hgb: S1 =     5 0 0 5 , S2 =     0 0 0 0 , S3 =       a 0 0 0 a 0 0 0 a s4 =       7 0 0 0 7 0 0 0 7 cÖ‡Z ̈KwU † ̄‹jvi g ̈vwUa·| `aóe ̈: mKj † ̄‹jvi g ̈vwUa·B KY© g ̈vwUa·| A‡f`K g ̈vwUa·/GKK g ̈vwUa· (Identity or Unit Matrix): KY© g ̈vwUa‡·i gyL ̈KY©w ̄’Z mKj fyw3 1 n‡j, Zv‡K A‡f`K g ̈vwU· e‡j| A‡f`K g ̈vwUa·‡K I Øviv m~wPZ Kiv nq| `aóe ̈: mKj A‡f`K g ̈vwUa·B † ̄‹jvi Ges KY© g ̈vwUa·| k~b ̈ g ̈vwUa· (zero matrix): †h g ̈vwUa‡·i mKj fzw3 k~b ̈ Zv‡K k~b ̈ g ̈vwUa· e‡j| †hgb:     0 0 0 0 ,       0 0 0 0 0 0 0 0 0 cÖ‡Z ̈KwUB k~b ̈ g ̈vwUa· m  n AvKv‡ii k~b ̈ g ̈vwUa·‡K 0m, n Øviv m~wPZ Kiv nq| †hgb: 01, 2 = [ 0 0 ] 02,3 =     0 0 0 0 0 0
4 24. A =       4 0 2 x  – 2 0 2 – 7 , B =       4 1 5 1 1 8 – 3 y – ; C = A – B GKwU wecÖwZmg g ̈vwUa· n‡j, (x, y) KZ? [A =       4 0 2 x  – 2 0 2 – 7 , B =       4 1 5 1 1 8 – 3 y – ; If C = A – B is an skew-symmetric matrix, then (x,y) =?] (2, – 8) (2, 4)  (2, – 4) DËi: (2, – 8) e ̈vL ̈v: A – B =       0 – 1 – 3 x – 1 0 – 10 3 2 – y 0 wecÖwZmg g ̈vwUa· n‡Z n‡j, x – 1 = 1 Ges 2 – y = 10 x = 2, y = – 8  (x, y) = (2, – 8) 25.       2 3 1 0 0 0 – 4 0 – 2 GKwU k~b ̈NvZx g ̈vwUa· n‡j, k~b ̈Nv‡Zi m~PK KZ? [       2 3 1 0 0 0 – 4 0 – 2 If it is a nilpotent matrix, what is the index of zero exponent?] 0 2 3 4 DËi: 3 e ̈vL ̈v:       2 3 1 0 0 0 – 4 0 – 2 2 =       0 6 0 0 0 0 0 – 12 0 Ges       2 3 1 0 0 0 – 4 0 – 2 3 =       0 0 0 0 0 0 0 0 0 26. 3A – B =     5 1 0 1 I B =     4 2 3 5 n‡j, A = ? [3A – B =     5 1 0 1 and B =     4 2 3 5 then, A = ?]     3 1 1 2     1 3 2 1     1 1 2 3     2 3 1 1 DËi:     3 1 1 2 e ̈vL ̈v: 3A – B =     5 1 0 1  3A =     5 1 0 1 + B =     5 1 0 1 +     4 2 3 5 =     9 3 3 6  A = 1 3     9 3 3 6 =     3 1 1 2 27. A =     3 6 9 0 , B =     9 4 3 5 n‡j, A T + BT = ? [A =     3 6 9 0 , B =     9 4 3 5 then, AT + BT = ?]     12 12 10 0     0 10 12 0     12 12 10 3     12 12 10 5 DËi:     12 12 10 5 e ̈vL ̈v: A T =     3 9 6 0 B T =     9 3 4 5 A T + BT =     12 12 10 5 28. A =     0 0 – 1 2 I B =     3 0 5 0 n‡j, |AB| Gi gvb KZ? [A =     0 0 – 1 2 and B =     3 0 5 0 then,what is value of |AB|?] 0 1 2 3 DËi: 0 e ̈vL ̈v: AB =     0 0 0 0  |AB| = 0 29. AB = C n‡j, wb‡Pi †KvbwU mwVK? [AB=C then, Which of the following is correct?] A = CB–1 B = A–1C B = C–1A K I L DfqB DËi: K I L DfqB e ̈vL ̈v: AB = C  A –1AB = A–1C  B = A–1C AB = C  AB.B–1 = CB–1  A = CB–1 30. A =      1 0 1 , B =     1 2 0 1 Ges A 2 = B n‡j,  Gi gvb- [A =      1 0 1 , B =     1 2 0 1 and A 2 = B then, what is value of α?]  2  1 1 – 1 DËi:  1 e ̈vL ̈v: A 2 = B       1 0 1      1 0 1 =     1 2 0 1       2  + 1 0 1 =     1 2 0 1  2 = 1   =  1 ---