Matrices and Determinants — practice questions

114 curated practice questions from Matrices and Determinants (Mathematics JEE Advanced, NTA Exams) — 64 hard, 50 medium. Every question includes the final answer free; step-by-step worked solutions with a free account.

• Assertion : | array ccc ( + ) & ( + ) & ( + ) ( + ) & ( + ) & ( + ) ( - ) & ( - ) & ( - ) array | is independent of θ…
• Let 0 (x, )= | array ccc x & & - & -x & 1 & 1 & x array | then
• If A= [ array ccc 1 & 0 & 0 0 & 1 & 1 0 & -2 & 4 array ] , 6 A –1 = A 2 + c A + d I , then ( c, d ) is
• If ω ( ≠ 1) is a cube root of unity, then | array ccc 1 & 1+i+ ^ 2 & ^ 2 1-i & -1 & ^ 2 -1 -i & -i+ -1 & -1 array |=
• Let M and N be two 3 × 3 matrices such that MN = NM . Further, if M ≠ N 2 and M 2 = N 4 , then
• If A= [ array cc & 0 1 & 1 array ] and B= [ array ll 1 & 0 5 & 1 array ], then value of α for which A 2 = B , is
• D = | array lll a & 1 & 1 1 & b & 1 1 & 1 & c array | = 0, then the value of 1 1-a + 1 1-b + 1 1-c is :
• If a, b, c > 0 & x, y, z ∈ R then the determinant | array ccc (a^ x +a^ -x )^ 2 & (a^ x -a^ -x )^ 2 & 1 (b^ y +b^ -y )^…
• Let ∆ = 0 and ∆ c denotes the determinant of cofactors, then ∆ c = ∆ n – 1 , where n (> 0) is the order of ∆. On the…
• Let S be the set of all column matrices [ array l b_ 1 b_ 2 b_ 3 array ] such that b 1 , b 2 , b 3 ∈ R and the system…
• Suppose X = ( array ll a & ~b c & ~d array ) satisfies the equation X 2 – 4X + 3I = O. Assertion : If a + d ≠ 4, then…
• Let P _ 4 = [ array lll 0 & 1 & 0 0 & 0 & 1 1 & 0 & 0 array ], P _ 5 = [ array lll 0 & 0 & 1 1 & 0 & 0 0 & 1 & 0 array…
• Let p be an odd prime number and T p be the following set of 2 × 2 matrices : The number of A in T p such that det ( A…
• Let M be a 2 × 2 symmetric matrix with integer entries. Then M is invertible if
• If a 2 + b 2 + c 2 = 1, then | array ccc a^ 2 + (b^ 2 +c^ 2 ) & b(1- ) & a(1- ) a(1- ) & b^ 2 + (c^ 2 +a^ 2 ) & b(1- )…
• Assertion : | array lll a ^ 2 + X ^ 2 & b - z & a + k b + I & b ^ 2 + X ^ 2 & b - x a - k & b + x & c ^ 2 + X ^ 2 array…
• Suppose D = | array lll a _ 1 & ~b _ 1 & c _ 1 a _ 2 & ~b _ 2 & c _ 2 a _ 3 & ~b _ 3 & c _ 3 array | and D’ = | array…
• | array ccc 1 & a & a^ 2 (z-z) & z & (z-z) (z-z) & z & (z-z) array | depends on
• Let ω ≠ 1 be a cube root of unity and S be the set of all non-singular matrices of the form [ array ccc 1 & a & b & 1 &…
• If ∆ = | array lll 1 & a & ~b 1 & ~b & a 1 & c & ~b array | = | array lll 1 & a & a ^ 2 1 & ~b & ~b ^ 2 1 & c & c ^ 2…
• Consider the set A of all determinants of order 3 with entries 0 or 1. Let B be the subset of A consisting of all…
• Let f (x) = | array ccc 1 / x & ~h x & x ^ n 1 & -1 / n & (-1)^ n 1 & a & a ^ 2 array | , then d^ n d^ n f (x) at x = 1…
• If E(θ) = [ array cc ^ 2 & & ^ 2 array ] and θ and φ differ by an odd multiple of π/2, then E(θ). E(φ) is a
• Let A, B and C be 2 × 2 matrices with entries from the set of real numbers. Define * as follows : A ^ * ~B = 1 2 ( AB ^…
• How many 3 × 3 matrices M with entries from 0, 1, 2 are there, for which the sum of the diagonal entries M T M is 5?
• If | array ccc x^ 2 +x & x+1 & x-2 2 x^ 2 +3 x-1 & 3 x & 3 x-3 x^ 2 +2 x+3 & 2 x-1 & 2 x-1 array | = Ax + B, where A…
• If a 2 + b 2 + c 2 = –2 and f (x) = Then f (x) is a polynomial of degree :
• Let a i , b i , c i ∈ N for i = 1, 2, 3 and let = | array lll 1+a_ 1 ^ 3 b_ 1 ^ 3 1+a_ 1 b_ 1 & 1+a_ 1 ^ 3 b_ 2 ^ 3…
• 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…
• 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…
• The system of equations x + 2y + 3z = 4, 2x + 3y + 4z = 5, 3x + 4y + 5z = 6 has
• If U n = | array ccc n & 1 & 5 n ^ 2 & 2 ~N +1 & 2 ~N +1 n ^ 3 & 3 ~N ^ 2 & 3 ~N +1 array | , then _ n=1 ^ N U_ n is…
• If x N and ^ x C _ i , ^ x ^ 2 C _ i and x^ 3 C_ i , (i = 1, 2, 3) are binomial coefficients, then 12 | array ccc ^ x C…
• If A = diag (d 1 , d 2 , d 3 , ........ d n ), then A n is equal to
• Let | M | denote the determinant of a square matrix M . Let g: [0, 2 ] R be the function defined by g( heta)= f(…
• Let x ∈ R and let P= [ array lll 1 & 1 & 1 0 & 2 & 2 0 & 0 & 3 array ], Q= [ array ccc 2 & x & x 0 & 4 & 0 x & x & 6…
• Elementary Transformation of a matrix : The following operation on a matrix are called elementary operations…
• Let ∆ = 0 and ∆ c denotes the determinant of cofactors, then ∆ c = ∆ n – 1 , where n (> 0) is the order of ∆. On the…
• Let ∆ = 0 and ∆ c denotes the determinant of cofactors, then ∆ c = ∆ n – 1 , where n (> 0) is the order of ∆. On the…
• If A = [ array cc 1 & x - x & 1 array ] , then the value of |A T A –1 | is
• For what value of x, the matrix [ array ccc 3-x & 2 & 2 2 & 4-x & 1 -2 & -4 & -1-x array ] is singular.
• The value of θ lying between θ = 0 and = 2 and satisfying the equation | array ccc 1+ ^ 2 & ^ 2 & 4 4 ^ 2 & 1+ ^ 2 & 4…
• Elementary Transformation of a matrix : The following operation on a matrix are called elementary operations…
• The parameter on which the value of the determinant does not depend upon
• If and | A 3 |= 125, then the value of α is
• If | array ccc x ^ 2 -2 x +3 & 7 x +2 & x +4 2 x +7 & x ^ 2 - x +2 & 3 x 3 & 2 x -1 & x ^ 2 -4 x +7 array | = ax 6 + bx…
• If A = [ array cc 4 & x+2 2 x-3 & x+1 array ] is symmetric, then x =
• if U 1 , U 2 and U 3 are columns matrices satisfying and U is 3 × 3 matrix when columns are U 1 , U 2 , U 3 then answer…
• Consider the system of equations x – 2 y + 3 z = – 1, – x + y – 2 z = k, x – 3 y + 4 z = 1 . Statement-1 : The system…
• If | array ccc x & 3 & 6 3 & 6 & x 6 & x & 3 array | = | array ccc 2 & x & 7 x & 7 & 2 7 & 2 & x array | = | array ccc…
• Elementary Transformation of a matrix : The following operation on a matrix are called elementary operations…
• If = | array ccc e ^ x & x & 1 x & _ e (1+ x ^ 2 ) & 1 x & x ^ 2 & 1 array |= a + bx + cx ^ 2 then
• The no. of distinct real roots of | array ccc x & x & x x & x & x x & x & x array |=0 in the interval is
• If the system of equations x + ay = 0, az + y = 0 and ax + z = 0 has infinitely many solutions, then the value of a is
• If A and B are square matrices of equal degree, then which one is correct among the following ?
• If m is a positive integer and ∆ r = | array ccc 2 r-1 & ^ m C_ r & 1 m^ 2 -1 & 2^ m & m+1 ^ 2 (m^ 2 ) & ^ 2 (m) & ^ 2…
• The determinant | array ccc x p+y & x & y y p+z & y & z 0 & x p+y & y p+z array |=0 , if
• Let where α ∈ R . Suppose Q = [ q ij ] is a matrix such that PQ = kI , where k ∈ R , k ≠ 0 and I is the identity matrix…
• If the system of linear equations x + 2ay + az = 0 x + 3by + bz = 0 x + 4cy + cz = 0 has a non-zero solution, then a…
• For any 3 × 3 matrix M , let | M | denote the determinant of M . Let I be the 3 × 3 identity matrix. Let E and F be two…
• If F (x) = [ array ccc x & - x & 0 x & x & 0 0 & 0 & 1 array ] and G (y) = [ array ccc y & 0 & y 0 & 1 & 0 - y & 0 & y…
• Let A( )= ( array cc + & 2 - 2 & - array ) Assertion : [A (π(3)] 3 = –I Reason : A (θ) A (φ) = A (θ + φ)
• Let f 1 (x) = x + a, f 2 (x) = x 2 + bx + c and ∆ = | array ccc 1 & 1 & 1 f _ 1 ( x _ 1 ) & f _ 1 ( x _ 2 ) & f _ 1 ( x…
• The system of equation –2x + y + z = 1, x –2y + z = –2, x + y + λz = 4 will have no solution if
• If the adjoint of a 3 × 3 matrix P is [ array lll 1 & 4 & 4 2 & 1 & 7 1 & 1 & 3 array ], then the possible value(s) of…
• a, b, c are non-zero real numbers. Then | array lll c & a & ~b a & ~b & ~b ~b & ~b & a array | = 0, if
• If P= [ array cc 3 2 & 1 2 - 1 2 & 3 2 array ], A= [ array ll 1 & 1 0 & 1 array ] and Q = PAP T , then P T ( Q 2005 ) P…
• If f(θ) = | array ccc 1 & 1 & -1 1 & e^ i & 1 1 & -1 & -e^ -i array | then
• The determinant | array ccc a & b & a +b b & c & b +c a +b & b +c & 0 array | is equal to zero, if
• Let ∆ = 0 and ∆ c denotes the determinant of cofactors, then ∆ c = ∆ n – 1 , where n (> 0) is the order of ∆. On the…
• If a > b > c and the system of equations ax + by + cz = 0, bx + cy + az = 0, cx + ay + bz = 0 has a non trivial…
• Suppose a, b, c are distinct real numbers. Let f(x)= | array lll x^ 3 +a^ 3 & x^ 2 -a^ 2 & x+a x^ 3 +b^ 3 & x^ 2 -b^ 2…
• Let D r = | array ccc a & 2^ r & 2^ 16 -1 b & 3 (4^ r ) & 2 (4^ 16 -1 ) c & 7 (8^ r ) & 4 (8^ 16 -1 ) array | , then…
• Let M be a 3 × 3 invertible matrix with real entries and let I denote the 3 × 3 identity matrix. If M −1 = adj(adj M )…
• Let ∆ = 0 and ∆ c denotes the determinant of cofactors, then ∆ c = ∆ n – 1 , where n (> 0) is the order of ∆. On the…
• If f(x) and g(x) are functions such that f (x + y) = f(x) g(y) + g(x) f (y), then | array lll f( ) & g( ) & f( + ) f( )…
• Let A and B be two matrices different from I such that AB = BA and A n – B n is invertible for some positive integer n…
• Consider the matrix P = ( array lll 2 & 0 & 0 0 & 2 & 0 0 & 0 & 3 array ) . Let the transpose of a matrix X be denoted…
• Given 2 x – y – z = 2, x – 2 y + z = –4, x + y + λ z = 4 then the value of λ such that the given system of equation has…
• Let a , b , c be the real numbers. Then following system of equations in x , y and z has
• The determinant ∆ = | array ccc a^ 2 +x^ 2 & b & a b & b^ 2 +x^ 2 & b a & b & c^ 2 +x^ 2 array | is divisible by
• For any 3 × 3 matrix M , let | M | denote the determinant of M . Let E= [ array ccc 1 & 2 & 3 2 & 3 & 4 8 & 13 & 18…
• Let f (a, b) = , then
• Let ax 7 + bx 6 + cx 5 + dx 4 + ex 3 + fx 2 + gx + h = | array ccc ( x +1) & ( x ^ 2 +2 ) & ( x ^ 2 + x ) ( x ^ 2 + x )…
• Which of the following is (are) NOT the square of a 3 × 3 matrix with real entries?
• Assertion : The system of equations possess a non trivial solution over the set of rationals x + ky + 3z = 0, 3x + ky –…
• Let a , λ, μ ∈ R . Consider the system of linear equations ax + 2 y = λ, 3 x – 2 y = μ. Which of the following…
• The number of 3 × 3 matrices A whose entries are either 0 or 1 and for which the system A [ array l x y z array ]= […
• The system of equations αx + y + z = α – 1 x + αy + z = α – 1 x + y + αz = α –1 has no solution, if α is :
• The value of the determinant | array ccc 6 & 2 i & 3+ 6 2 & 3 + 8 i & 3 2 + 6 i 8 & 2 + 2 i & 2 +2 i array | , where i…
• Let M= [ array lll 0 & 1 & a 1 & 2 & 3 3 & b & 1 array ] and adj M= [ array ccc -1 & 1 & -1 8 & -6 & 2 -5 & 3 & -1…
• The digits A, B, C are such that the three digit numbers A 88, 6B8, 86C are divisible by 72, then the determinant |…
• If P is a 3 × 3 matrix such that P T = 2 P + I , where P T is the transpose of P and I is the 3 × 3 identity matrix…
• If f (x) = | array ccc 1+ ^ 2 x & ^ 2 x & 4 2 x ^ 2 x & 1+ ^ 2 x & 4 2 x ^ 2 x & ^ 2 x & 1+4 2 x array | , f (a ) and f…
• Let X and Y be two arbitrary, 3 × 3 non-zero, skew-symmetric matrices and Z be an arbitrary 3 × 3, non-zero, symmetric…
• Let M= [ array cc ^ 4 & -1- ^ 2 1+ ^ 2 & ^ 4 array ]= I+ M^ -1 where α = α(θ) and β = β(θ) are real numbers, and I is…
• Which of the following values of α satisfy the equation | array ccc (1+ )^ 2 & (1+2 )^ 2 & (1+3 )^ 2 (2+ )^ 2 & (2+2 )^…
• Let A be the set of all 3 × 3 symmetric matrices all of whose entries are either 0 or 1 . Five of these entries are 1…
• If f(x)= | array ccc 2 x & 2 x & 2 x - x & x & - x x & x & x array | , then
• If f ( x ) = | array ccc 1 & x & x+1 2 x & x(x-1) & (x+1) x 3 x(x-1) & x(x-1)(x-2) & (x+1) x(x-1) array | then f (100)…
• If f(x), g (x) and h (x) are three polynomials of degree 2, then φ (x) = | array ccc f(x) & g(x) & h(x) f^ (x) & g^ (x)…
• If n is not a multiple of 3 and 1, ω, ω 2 are the cube roots of unity, then ∆ = | array ccc 1 & ^ n & ^ 2 n ^ 2 n & 1 &…
• Let M = ( a ij ), i , j ∈ 1, 2, 3 , be the 3 × 3 matrix such that a ij = 1 if j + 1 is divisible by i , otherwise a ij…
• Let A be a symmetric matrix such that A 5 = O and B = I + A + A 2 + A 3 + A 4 , then B is
• For 3 × 3 matrices M and N , which of the following statement(s) is (are) not correct?
• ∆ = | array ccc p & 2- i & i + l 2+ i & q & 3+ i 1- i & 3- i & r array | is always (where p,q,r ∈ R)
• Let P= [ array ccc 1 & 0 & 0 4 & 1 & 0 16 & 4 & 1 array ] and I be the identity matrix of order 3. If Q = [ q ij ] is a…
• Assertion : If ∆ (x) = | array ll f _ 1 ( x ) & f _ 2 ( x ) g _ 1 ( x ) & g _ 2 ( x ) array | , then ∆’ (x) ≠ | array…
• Assertion : f (x) = array lll (1+x)^ 1 & (1+x)^ 2 & (1+c)^ 3 (1+x)^ 3 & (1+x)^ 3 & (1+x)^ 3 (1+x)^ 4 & (1+x)^ 4 &…
• Let I= ( array ll 1 & 0 0 & 1 array ) and P= ( array ll 2 & 0 0 & 3 array ) . Let Q= ( array ll x & y z & 4 array ) for…
• Eliminating a, b, c from x= a b-c , y= b c-a , z= c a-b , we get
• 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 …
• Which one of the following is correct ? If A is non-singular matrix, then, :
• If | array ccc p & q-y & r-z p-x & q & r-z p-x & q-y & r array | = 0, then the value of p x + q y + r z is

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