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JEE Main Matrices and Determinants questions

70 Mathematics questions on Matrices and Determinants as asked in JEE Main, newest first. 35 come with a worked solution.

  1. Let A = beginbmatrix frac1√(10) & frac3√(10) \ frac-3√(10) & frac1√(10) endbmatrix and B = beginbmatrix 1 & -i \ 0 & 1 endbmatrix, where i = √(-1). If M = A^T B A, then the inverse of the matrix AM^2023A^T is

    2023

  2. Let S_1 and S_2 be respectively the sets of all a ∈ R - \0\ for which the system of linear equations ax + 2ay - 3az = 1 (2a+1)x + (2a+3)y + (a+1)z = 2 (3a+5)x + (a+5)y + (a+2)z = 3 has unique solution and infinitely…

    2023

  3. Let x, y, z > 1 and A = beginbmatrix 1 & log_x y & log_x z \ log_y x & 2 & log_y z \ log_z x & log_z y & 3 endbmatrix. Then |adj(adj A^2)| is equal to

    2023

  4. Let A, B, C be 3 × 3 matrices such that A is symmetric and B and C are skew-symmetric. Consider the statements (S1) A^13 B^26 - B^26 A^13 is symmetric (S2) A^26 C^13 - C^13 A^26 is symmetric Then,

    2023 · with solution

  5. If the system of equations x + 2y + 3z = 3 4x + 3y - 4z = 4 8x + 4y - λ z = 9 + μ has infinitely many solutions, then the ordered pair (λ, μ) is equal to :

    2023 · with solution

  6. Consider the following system of equations ax + 2y + z = 1 2ax + 3y + z = 1 3x + ay + 2z = β for some α, β ∈ R. Then which of the following is NOT correct.

    2023

  7. Let α and β be real numbers. Consider a 3 × 3 matrix A such that A^2 = 3A + α I. If A^4 = 21A + β I, then

    2023 · with solution

  8. If A and B are two non-zero n × n matrices such that A^2 + B = A^2 B, then

    2023

  9. Let A be a 3 × 3 matrix such that |adj(adj(adj A))| = 12^4. Then |A^-1 adj A| is equal to

    2023

  10. A value of θ ∈ (0, π/3), for which beginvmatrix 1 + cos^2θ & sin^2θ & 4cos 6θ \ cos^2θ & 1 + sin^2θ & 4cos 6θ \ cos^2θ & sin^2θ & 1 + 4cos 6θ endvmatrix = 0 is

    2019 · with solution

  11. If [x] denotes the greatest integer ≤ x, then the system of liner equations [sinθ]x + [-cosθ]y = 0, [cotθ]x + y = 0

    2019

  12. If B = beginbmatrix 5 & 2α & 1 \ 0 & 2 & 1 \ α & 3 & -1 endbmatrix is the inverse of a 3 × 3 matrix A, then the sum of all values of α for which det(A) + 1 = 0, is

    2019

  13. If A is a symmetric matrix and B is a skew-symmetric matrix such that A + B = beginbmatrix 2 & 3 \ 5 & -1 endbmatrix, then AB is equal to

    2019 · with solution

  14. The sum of the real roots of the equation beginvmatrix x & -6 & -1 \ 2 & -3x & x-3 \ -3 & 2x & x+2 endvmatrix = 0, is equal to

    2019

  15. The total number of matrices A = beginbmatrix 0 & 2y & 1 \ 2x & y & -1 \ 2x & -y & 1 endbmatrix, (x, y ∈ R, x ≠ y) for which A^T A = 3I_3 is

    2019

  16. Let λ be a real number for which the system of linear equations x + y + z = 6, 4x + λ y - λ z = λ - 2 and 3x + 2y - 4z = -5 has infinitely many solutions. Then λ is a root of the quadratic equation

    2019 · with solution

  17. Let A = beginbmatrix cosα & -sinα \ sinα & cosα endbmatrix (α ∈ R) such that A^32 = beginbmatrix 0 & -1 \ 1 & 0 endbmatrix. Then, a value of α is

    2019

  18. If Δ_1 = beginvmatrix x & sinθ & cosθ \ -sinθ & -x & 1 \ cosθ & 1 & x endvmatrix and Δ_2 = beginvmatrix x & sin 2θ & cos 2θ \ -sin 2θ & -x & 1 \ cos 2θ & 1 & x endvmatrix, x ≠ 0, then for all θ ∈ (0, (π)/2 )

    2019 · with solution

  19. If the system of linear equations x + y + z = 5 x + 2y + 2z = 6 x + 3y + λ z = μ, (λ, μ ∈ R), has infinitely many solutions, then the value of λ + μ is

    2019 · with solution

  20. Let A and B be two invertible matrices of order 3 × 3. If det(ABA^T) = 8 and det(AB^-1) = 8, then det(BA^-1B^T) is equal to

    2019 · with solution

  21. Let P = beginbmatrix 1 & 0 & 0 \ 3 & 1 & 0 \ 9 & 3 & 1 endbmatrix and Q = [q_ij] be two 3 × 3 matrices such that Q - P^3 = I_3. Then, dfracq_21 + q_31q_32 is equal to

    2019

  22. If the system of equations 2x + 3y - z = 0, x + ky - 2z = 0 and 2x - y + z = 0 has a non-trivial solution (x, y, z), then x/y + y/z + z/x + k is equal to

    2019

  23. If beginbmatrix 1 & 1 \ 0 & 1 endbmatrix beginbmatrix 1 & 2 \ 0 & 1 endbmatrix beginbmatrix 1 & 3 \ 0 & 1 endbmatrix ·s beginbmatrix 1 & n-1 \ 0 & 1 endbmatrix = beginbmatrix 1 & 78 \ 0 & 1 endbmatrix, then the inverse…

    2019 · with solution

  24. If A = beginbmatrix cosθ & -sinθ \ sinθ & cosθ endbmatrix, then the matrix A^-50 when θ = (π)/12, is equal to

    2019

  25. If the system of linear equations x - 2y + kz = 1, 2x + y + z = 2, 3x - y - kz = 3 has a solution (x, y, z), z ≠ 0, then (x, y) lies on the straight line whose equation is

    2019

  26. Let A = beginbmatrix 0 & 2q & r \ p & q & -r \ p & -q & r endbmatrix. If AA^T = I_3, then |p| is

    2019 · with solution

  27. Let α and β be the roots of the equation x^2 + x + 1 = 0. Then, for y ≠ 0 in R, beginvmatrix y+1 & α & β \ α & y+β & 1 \ β & 1 & y+α endvmatrix is equal to

    2019

  28. Let the numbers 2, b, c be in an AP and A = beginbmatrix 1 & 1 & 1 \ 2 & b & c \ 4 & b^2 & c^2 endbmatrix If det(A) ∈ [2, 16], then c lies in the interval

    2019 · with solution

  29. The set of all values of λ for which the system of linear equations x - 2y - 2z = λ x x + 2y + z = λ y and -x - y = λ z has a non-trivial solution

    2019

  30. If beginvmatrix a-b-c & 2a & 2a \ 2b & b-c-a & 2b \ 2c & 2c & c-a-b endvmatrix = (a+b+c)(x+a+b+c)^2, x ≠ 0 and a+b+c ≠ 0, then x is equal to

    2019

  31. An ordered pair (α, β) for which the system of linear equations (1 + α)x + β y + z = 2 α x + (1 + β)y + z = 3 α x + β y + 2z = 2 has a unique solution, is

    2019

  32. If the system of linear equations 2x + 2y + 3z = a 3x - y + 5z = b x - 3y + 2z = c where a, b, c are non-zero real numbers, has more than one solution, then

    2019 · with solution

  33. Let A = beginbmatrix 2 & b & 1 \ b & b^2+1 & b \ 1 & b & 2 endbmatrix, where b > 0. Then, the minimum value of (det(A))/b is

    2019 · with solution

  34. The number of values of θ ∈ (0, π) for which the system of linear equations x + 3y + 7z = 0, -x + 4y + 7z = 0, (sin 3θ)x + (cos 2θ)y + 2z = 0 has a non-trivial solution, is

    2019

  35. Let d ∈ R, and A = beginbmatrix -2 & 4+d & (sinθ)-2 \ 1 & (sinθ)+2 & d \ 5 & (2sinθ)-d & (-sinθ)+2+2d endbmatrix, θ ∈ [0, 2π]. If the minimum value of det(A) is 8, then a value of d is

    2019

  36. If the system of equations x + y + z = 5, x + 2y + 3z = 9, x + 3y + α z = β has infinitely many solutions, then β - α equals

    2019 · with solution

  37. If the system of linear equations x - 4y + 7z = g, 3y - 5z = h, -2x + 5y - 9z = k is consistent, then

    2019 · with solution

  38. The system of linear equations x + y + z = 2, 2x + 3y + 2z = 5 2x + 3y + (a^2 - 1)z = a + 1

    2019 · with solution

  39. If beginvmatrix x-4 & 2x & 2x \ 2x & x-4 & 2x \ 2x & 2x & x-4 endvmatrix = (A + Bx)(x - A)^2, then the ordered pair (A, B) is equal to

    2018 · with solution

  40. If the system of linear equations x + ky + 3z = 0, 3x + ky - 2z = 0 2x + 4y - 3z = 0 has a non-zero solution (x, y, z), then xz/(y^2) is equal to

    2018

  41. If A = beginbmatrix 2 & -3 \ -4 & 1 endbmatrix, then adj(3A^2 + 12A) is equal to

    2017 · with solution

  42. Let ω be a complex number such that 2ω + 1 = z, where z = √(-3). If beginvmatrix 1 & 1 & 1 \ 1 & -ω^2 - 1 & ω^2 \ 1 & ω^2 & ω^7 endvmatrix = 3k, then k is equal to

    2017

  43. If A = beginbmatrix 5a & -b \ 3 & 2 endbmatrix and A adj A = AA^T, then 5a + b is equal to

    2016 · with solution

  44. Let P = beginbmatrix 1 & 0 & 0 \ 4 & 1 & 0 \ 16 & 4 & 1 endbmatrix and I be the identity matrix of order 3. If Q = [q_ij] is a matrix, such that P^50 - Q = I, then dfracq_31 + q_32q_21 equals

    2016

  45. The system of linear equations x + λ y - z = 0; λ x - y - z = 0; x + y - λ z = 0 has a non-trivial solution for

    2016

  46. If A = beginbmatrix 1 & 2 & 2 \ 2 & 1 & -2 \ c & 2 & b endbmatrix is a matrix satisfying the equation AA^T = 9I, where I is 3 × 3 identity matrix, then the ordered pair (a, b) is equal to

    2015

  47. The set of all values of λ for which the system of linear equations 2x_1 - 2x_2 + x_3 = λ x_1, 2x_1 - 3x_2 + 2x_3 = λ x_2, -x_1 + 2x_2 = λ x_3 has a non-trivial solution

    2015

  48. If A is a 3 × 3 non-singular matrix such that AA^T = A^T A and B = A^-1A^T, then BB^T is equal to

    2014 · with solution

  49. If α, β ≠ 0 and f(n) = α^n + β^n and beginvmatrix 3 & 1+f(1) & 1+f(2) \ 1+f(1) & 1+f(2) & 1+f(3) \ 1+f(2) & 1+f(3) & 1+f(4) endvmatrix = K(1-α)^2(1-β)^2(α-β)^2, then K is equal to

    2014 · with solution

  50. If P = beginbmatrix 1 & α & 3 \ 1 & 3 & 3 \ 2 & 4 & 4 endbmatrix is the adjoint of a 3 × 3 matrix A and |A| = 4, then α is equal to

    2013 · with solution

  51. The number of value of k, for which the system of equation (k+1)x + 8y = 4y ⇒ kx + (k+3)y = 3k - 1 has no solution, is

    2013

  52. If P is a 3 × 3 matrix such that P^T = 2P + I, where P^T is the transpose of P and I is the 3 × 3 identity matrix, then there exists a column matrix, X = beginbmatrix x \ y \ z endbmatrix ≠ beginbmatrix 0 \ 0 \ 0…

    2012 · with solution

  53. Let P = [a_ij] be a 3 × 3 matrix and let Q = [b_ij], where b_ij = 2^i+j a_ij for 1 ≤ i, j ≤ 3. If the determinant of P is 2, then the determinant of the matrix Q is

    2012 · with solution

  54. Let M and N be two 3 × 3 non-singular skew-symmetric matrices such that MN = NM. If P^T denotes the transpose of P, then M^2N^2(M^TN)^-1(MN^-1)^T is equal to

    2011 · with solution

  55. The number of 3 × 3 matrices A whose entries are either 0 or 1 and for which the system A beginbmatrix x \ y \ z endbmatrix = beginbmatrix 1 \ 0 \ 0 endbmatrix has exactly two distinct solutions, is

    2010

  56. If P = beginbmatrix √(3)/2 & 1/2 \ -1/2 & √(3)/2 endbmatrix, A = beginbmatrix 1 & 1 \ 0 & 1 endbmatrix and Q = PAP^T, then P^TQ^2005P is

    2005 · with solution

  57. If A = beginbmatrix 1 & 0 & 0 \ 0 & 1 & 1 \ 0 & -2 & 4 endbmatrix, 6A^-1 = A^2 + cA + dI, then (c, d) is

    2005 · with solution

  58. If A = beginbmatrix α & 2 \ 2 & α endbmatrix and |A^3| = 125, then the value of α is

    2004 · with solution

  59. Given, 2x - y + 2z = 2, x - 2y + z = -4, x + y + λ z = 4, then the value of λ such that the given system of equations has no solution, is

    2004

  60. If A = beginbmatrix α & 0 \ 1 & 1 endbmatrix and B = beginbmatrix 1 & 0 \ 5 & 1 endbmatrix then value of α for which A^2 = B, is

    2003 · with solution

  61. Let ω = -1/2 + idfrac√(3)2, then value of the determinant beginvmatrix 1 & 1 & 1 \ 1 & -1-ω^2 & ω^2 \ 1 & ω^2 & ω endvmatrix is

    2002 · with solution

  62. The number of values of k for which the system of equations (k+1)x + 8y = 4k and kx + (k+3)y = 3k - 1 has infinitely many solutions, is/are

    2002

  63. The number of distinct real roots of beginvmatrix sin x & cos x & cos x \ cos x & sin x & cos x \ cos x & cos x & sin x endvmatrix = 0 in the interval -(π)/4 ≤ x ≤ (π)/4 is

    2001 · with solution

  64. If the system of equations x - ky - z = 0, kx - y - z = 0, x + y - z = 0 has a non-zero solution, then possible values of k are

    2000

  65. If f(x) = beginvmatrix 1 & x & x+1 \ 2x & x(x-1) & (x+1)x \ 3x(x-1) & x(x-1)(x-2) & (x+1)x(x-1) endvmatrix, then f(100) is equal to

    1999

  66. The parameter on which the value of the determinant beginvmatrix 1 & a & a^2 \ cos(p-d)x & cos px & cos(p+d)x \ sin(p-d)x & sin px & sin(p+d)x endvmatrix does not depend upon, is

    1997

  67. The determinant beginvmatrix xp+y & x & y \ yp+z & y & z \ 0 & xp+y & yp+z endvmatrix = 0, if

    1997 · with solution

  68. If A and B are square matrices of equal degree, then which one is correct among the following?

    1995 · with solution

  69. Consider the set A of all determinants of order 3 with entries 0 or 1 only. Let B be the subset of A consisting of all determinants with value 1. Let C be the subset of A consisting of all determinants with value -1.…

    1981

  70. If the system of equations x + ay = 0, az + y = 0 and ax + z = 0 has infinite solutions, then the value of a is

    with solution

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