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Matrices and Determinants

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  1. 151.Among the statements: I: If vmatrix1&& &1& &&1vmatrix=0, then ^2+^2+^2=1, and II: If vmatr…
  2. 152.The system of linear equations [x+y+z=6, 2x+5y+az=b, x+2y+3z=6] has:
  3. 153.Let f(x)= 7x^10+9x^8(1+x^2+2x^9)^2,dx, x>0, with _x0f(x)=0 and f(1)=14.
  4. 154.Let P=[p_ij] and Q=[q_ij] be two square matrices of order 3 such that q_ij=2^(i+j-1)p_ij a…
  5. 155.Let A,B,C be three 22 matrices with real entries such that B=(I+A)^-1 and A+C=I.
  6. 156.Let A=pmatrix1&21&pmatrix and B=pmatrix3&3&2pmatrix.
  7. 157.Let ,R be such that the system of linear equations [x+2y+z=5, 2x+y+ z=5, 8x+4y+ z=18] has …
  8. 158.Let A=pmatrix1&1&2-2&0&11&3&5pmatrix.
  9. 159.Let S=A=pmatrixa&b&dpmatrix: a,b,c,d0,1,2,3,4, A^2-4A+3I=O.
  10. 160.Let A=pmatrix1&2&74&-2&83&8&-7pmatrix, and let be a real number satisfying (A- I)=0.
  11. 161.If the system of equations [x+y+z=5, x+2y+3z=9, x+3y+ z=] has infinitely many solutions, t…
  12. 162.Consider the system of linear equations in x,y,z: [x+2y+tz=0, 6x+y+5tz=0, 3x+t^2y+f(t)z=0,…
  13. 163.Let A be a 33 matrix such that [A^Tpmatrix101pmatrix=pmatrix522pmatrix, A^Tpmatrix001pmatr…
  14. 164.Let M be a 33 matrix such that Mpmatrix100pmatrix=pmatrix123pmatrix, Mpmatrix010pmatrix=pm…
  15. 165.If f:NZ is defined by [f(n)=_k=1^nvmatrixk&-1&-5 -2n^2 & 3(2k+1) & 2k+1 -3n^3 & 3k(2k+1) &…
  16. 166.Let A=pmatrix1&0&03&0&09&3&0pmatrix and B=I.
  17. 167.Let A=pmatrix&1&22&3&00&4&5pmatrix and B=pmatrix1&0&00&-5&00&4&-2pmatrix+adj(A).
  18. 168.If the system of linear equations [x+y+z=6, x+2y+5z=10, 2x+3y+ z=] has infinitely many sol…
  19. 169.Let A be a square matrix of order 3 such that (A)=-2 and (3adj(-6adj(3A)))=2^m+n3^mn, m>n.
  20. 170.Let A be a 33 matrix such that X^TAX=O for all nonzero 31 matrices X=pmatrixxpmatrix.
  21. 171.Let M denote the set of all real 33 matrices and S=-3,-2,-1,1,2.
  22. 172.Let S=mZ: A^m^2+A^m=3I-A^-6 where A=pmatrix2&-11&0pmatrix.
  23. 173.Let I be the identity matrix of order 3, and A=pmatrix&2&34&5&67&-1&2pmatrix with |A|=-1.
  24. 174.Let A=pmatrix&0&-0&1&0&0&pmatrix, (0,), and A^2=A^T.
  25. 175.For a 33 matrix M, let trace(M) denote the sum of its diagonal elements.
  26. 176.If A,B are non-singular matrices of the same order, then the inverse of A(adj(A^-1)+adj(B^…
  27. 177.If the system of equations [(-1)x+(-4)y+ z=5, x+(-1)y+(-4)z=7, (+1)x+(+2)y-(+2)z=9] has in…
  28. 178.Let A=[a_ij] be a 33 matrix such that Apmatrix010pmatrix=pmatrix010pmatrix, Apmatrix001pma…
  29. 179.The system of equations [5x+ y+3z=12, 2x-y+z=4, 100x-47y+ z=212] has no solution if:
  30. 180.If the system of equations [x+y+z=6, x+2y+5z=9, x+5y+ z=] has infinitely many solutions, t…
  31. 181.If the system of equations [2x-y+z=4, 5x+ y+3z=12, 100x-47y+ z=212] has infinitely many so…
  32. 182.For some a,b,R, let [f(x)=vmatrixa+ xx & 1 & b a & 1+ xx & b a & 1 & b+ xxvmatrix, x0, _x0…
  33. 183.Let A=pmatrix a & 1√2 1√2 & bpmatrix+1 (a 22 matrix built from real a,b), and P=pmatrix & …
  34. 184.Let M and m respectively be the maximum and the minimum values of [f(x)=vmatrix1+^2x & ^2x…
  35. 185.Let A=[a_ij]=pmatrix_5128 & _45 _58 & _425pmatrix.
  36. 186.Let A=[a_ij] be a 33 matrix of order 3, with a_ij=(√2)^i+j.
  37. 187.Let , () be the values of m for which the equations x+y+z=1, x+2y+4z=m, x+4y+10z=m^2 have …
  38. 188.Let A=[a_ij] be a 22 matrix such that a_ij0,1 for all i,j.
  39. 189.If the system of linear equations [3x+y+ z=3, 2x+ y-z=-3, x+2y+z=4] has infinitely many so…
  40. 190.Let A be a 33 matrix such that (A)=-4, and A+I=pmatrix1&a&12&1&0&1&2pmatrix, where I is th…
  41. 191.Let A=pmatrix & -16 & pmatrix, >0, such that (A)=0 and +=1.
  42. 192.Let A be a 33 real matrix such that A^2(A-2I)-4(A-I)=O, where I,O are the identity and nul…
  43. 193.Let A be a 33 matrix of order 3 with |A|=5.
  44. 194.Let the matrix A=pmatrix1&0&01&0&10&1&0pmatrix satisfy A^n=A^n-2+A^2-I for n3.
  45. 195.Let A be a 33 matrix such that |adj(adj(adjA))|=81.
  46. 196.Let A=pmatrix2&2+p&2+p+q4&6+2p&8+3p+2q6&12+3p&20+6p+3qpmatrix.
  47. 197.Let be a solution of x^2+x+1=0, and for some a,bR, pmatrix4&a&bpmatrixpmatrix1&16&13-1&-1&…
  48. 198.Let A=I_2-2MM^T, where M is a real matrix of order 21 such that M^TM=I_1.
  49. 199.If A and B are two non-zero n n matrices such that A^2+B=A^2B, then
  50. 200.The number of square matrices of order 5 with entries from the set 0,1, such that the sum …