Let A = [201110101], B = [B₁, B₂, B₃], where B₁, B₂, B₃ are column matrices, and AB₁ = [100]^T, AB₂ = [230]^T, AB₃ = [321]^T. If α = |B| and β is the sum of all the diagonal elements of B, then α³ + β³ is equal to ____.

Let A be a 2×2 real matrix and I be the identity matrix of order 2. If the roots of the equation |A − xI| = 0 be −1 and 3, then the sum of the diagonal elements of the matrix A² is ____.

The values of α, for which |13/2α+3/211/3α+1/32α+33α+10| = 0, lie in the interval.

Let A = [1000αβ0βα] and |2A|³ = 2²¹ where α, β ∈ ℤ. Then a value of α is ____.

Let for any three distinct consecutive terms a, b, c of an A.P, the lines ax + by + c = 0 be concurrent at the point P and Q(α, β) be a point such that the system of equations x + y + z = 6, 2x + 5y + α z = β and x + 2y + 3z = 4, has infinitely many solutions. Then (PQ)² is equal to ____.

Let A = [2126211332] and P = [120502715]. The sum of the prime factors of |P⁻¹AP − 2I| is equal to.

Let A be a square matrix such that AA^T = I. Then (1/2)A[(A + A^T)² + (A − A^T)²] is equal to.

Let R = [x000y000z] be a non-zero 3×3 matrix, where x sin θ = y sin(θ + 2π/3) = z sin(θ + 4π/3) ≠ 0, θ ∈ (0, 2π). For a square matrix M, let trace(M) denote the sum of all the diagonal entries of M. Then, among the statements: (I) Trace(R) = 0 (II) If trace(adj(adj(R))) = 0, then R has exactly one non-zero entry.

Consider the system of linear equations x + y + z = 5, x + 2y + λ² z = 9, x + 3y + λ z = μ, where λ, μ ∈ ℝ. Then, which of the following statement is NOT correct?

If the system of linear equations x − 2y + z = −4, 2x + α y + 3z = 5, 3x − y + β z = 3 has infinitely many solutions, then 12α + 13β is equal to.

Let A be a 3×3 matrix and det(A) = 2. If n = det(adj(adj(...(adj A)...))) with adj applied 2024 times, then the remainder when n is divided by 9 is equal to ____.

Let A be a 3×3 real matrix such that A(1,0,1)^T = 2(1,0,1)^T, A(−1,0,1)^T = 4(−1,0,1)^T, A(0,1,0)^T = 2(0,1,0)^T. Then, the system (A − 3I)(x,y,z)^T = (1,2,3)^T has ____.

If the system of equations 2x+3y−z=5, x+α y+3z=−4, 3x−y+β z=7 has infinitely many solutions, then 13 α β is equal to.

If A = [√21][−1√2], B = [10][11], C = ABA^T and X = A^T C² A, then det X is equal to:

Let A = I₂ − 2MM^T, where M is real matrix of order 2×1 such that the relation M^T M = I₁ holds. If λ is a real number such that the relation AX = λ X holds for some non-zero real matrix X of order 2×1, then the sum of squares of all possible values of λ is equal to:

Let the system of equations x + 2y + 3z = 5, 2x + 3y + z = 9, 4x + 3y + λ z = μ have infinite number of solutions. Then λ + 2μ is equal to:

Let A be a 3×3 matrix of non−negative real elements such that A[111]^T = 3[111]^T. Then the maximum value of det(A) is ____.

Let α ∈ (0, ∞) and A = [12α101012]. If det(adj(2A − A^T) . adj(A − 2A^T)) = 2⁸, then (det(A))² is equal to:

If the system of equations x + (√2 sin α)y + (√2 cos α)z = 0, x + (cos α)y + (sin α)z = 0, x + (sin α)y − (cos α)z = 0 has a non-trivial solution, then α ∈ (0, π/2) is equal to:

Let A = [1201] and B = I + adj(A) + (adj A)² + ... + (adj A)¹⁰. Then, the sum of all the elements of the matrix B is:

Let A be a 2×2 symmetric matrix such that A[11]^T = [37]^T and the determinant of A be 1. If A⁻¹ = α A + β I, where I is an identity matrix of order 2×2, then α + β equals ____.

Let A and B be two square matrices of order 3 such that |A| = 3 and |B| = 2. Then |A^T A (adj(2A))⁻¹ (adj(4B)) (adj(AB))⁻¹ A A^T| is equal to:

If the system of equations 11x + y + λ z = −5; 2x + 3y + 5z = 3; 8x − 19y − 39z = μ has infinitely many solutions, then λ⁴ − μ is equal to:

Let α, β ≠ 0 and A = [βα3ααβ−βα2α]. If B = [3α−93α−α7−2α−2α5−2β] is the matrix of cofactors of the elements of A, then det(AB) is equal to:

The values of m, n, for which the system of equations x + y + z = 4, 2x + 5y + 5z = 17, x + 2y + mz = n has infinitely many solutions, satisfy the equation:

Let α β γ = 45; α, β, γ ∈ ℝ. If x(α,1,2) + y(1,β,2) + z(2,3,γ) = (0,0,0) for some x, y, z ∈ ℝ, xyz ≠ 0, then 6α + 4β + γ is equal to ____.

For α, β ∈ ℝ and a natural number n, let A_r = |r1(n²+α)/22r2n²−β3r−23n(3n−1)/2|. Then 2A₁₀ − A₈ is ____.

Let A = [2−111]. If the sum of the diagonal elements of A¹³ is 3ⁿ, then n is equal to ____.

Let A = [2a013105b]. If A³ = 4A² − A − 21I, where I is the identity matrix of order 3×3, then 2a + 3b is equal to:

Let A be a non-singular matrix of order 3. If det(3 adj(2 adj((det A)A))) = 3⁻¹³ . 2⁻¹⁰ and det(3 adj(2A)) = 2^m . 3ⁿ, then |3m + 2n| is equal to ____.

Let λ, μ ∈ ℝ. If the system of equations 3x + 5y + λ z = 3, 7x + 11y − 9z = 2, 97x + 155y − 189z = μ has infinitely many solutions, then μ + 2λ is equal to:

Let B = [1315] and A be a 2×2 matrix such that AB⁻¹ = A⁻¹. If BCB⁻¹ = A and C⁴ + α C² + β I = 0, then 2β − α is equal to:

If the system of linear equations: x + y + 2z = 6, 2x + 3y + az = a + 1, −x − 3y + bz = 2b, where a, b ∈ ℝ, has infinitely many solutions, then 7a + 3b is equal to:

For a 3×3 matrix M, let trace(M) denote the sum of all the diagonal elements of M. Let A be a 3×3 matrix such that |A| = 1/2 and trace(A) = 3. If B = adj(adj(2A)), then the value of |B| + trace(B) equals:

If A, B, and (adj(A⁻¹) + adj(B⁻¹)) are non-singular matrices of same order, then the inverse of A(adj(A⁻¹) + adj(B⁻¹))⁻¹ B, is equal to.

If the system of equations (λ−1)x + (λ−4)y + λ z = 5, λ x + (λ−1)y + (λ−4)z = 7, (λ+1)x + (λ+2)y − (λ+2)z = 9 has infinitely many solutions, then λ² + λ is equal to.

The system of equations x + y + z = 6, x + 2y + 5z = 9, x + 5y + λ z = μ, has no solution if.

Let A = [a_ij] be a 3×3 matrix such that A(0,1,0)^T = (0,0,1)^T, A(4,1,3)^T = (0,1,0)^T and A(2,1,2)^T = (1,0,0)^T, then a₂₃ equals:

Let A be a 3×3 matrix such that X^T A X = O for all nonzero 3×1 matrices X = [xyz]^T. If A[111]^T = [14−5]^T, A[121]^T = [04−8]^T, and det(adj(2(A + I))) = 2^α 3^β 5^γ, α, β, γ ∈ ℕ, then α² + β² + γ² is ____.

If the system of equations 2x − y + z = 4, 5x + λ y + 3z = 12, 100x − 47y + μ z = 212 has infinitely many solutions, then μ − 2λ is equal to.

For some a, b, let f(x) = |a + sin x/x1ba1 + sin x/xba1b + sin x/x|, x ≠ 0, lim_x→0 f(x) = λ + μ a + ν b. Then (λ + μ + ν)² is equal to:

If the system of equations x + 2y − 3z = 2, 2x + λ y + 5z = 5, 14x + 3y + μ z = 33 has infinitely many solutions, then λ + μ is equal to:

Let M denote the set of all real matrices of order 3×3 and let S = {−3, −2, −1, 1, 2}. Let S₁ = {A = [a_ij] in M : A = A^T and a_ij in S, for all i, j}, S₂ = {A = [a_ij] in M : A = −A^T and a_ij in S, for all i, j}, S₃ = {A = [a_ij] in M : a₁₁ + a₂₂ + a₃₃ = 0 and a_ij in S, for all i, j}. If n(S₁ union S₂ union S₃) = 125 α, then α equals _____.

Let A = [1/√2−201] and P = [cos θ−sin θsin θcos θ], θ > 0. If B = P A P^T, C = P^T B¹⁰ P and the sum of the diagonal elements of C is m/n, where gcd(m, n) = 1, then m + n is:

Let S = {m ∈ ℤ : A^m2 + A^m = 3I − A⁶}, where A = [2−110]. Then n(S) is equal to ______.

Let A = [a_ij] = [log₅ 128log₄ 5log₅ 8log₄ 25]. If A_ij is the cofactor of a_ij, C_ij = Σ_k=1² a_ik A_jk, 1 ≤ i, j ≤ 2, and C = [C_ij], then 8|C| is equal to:

Let M and N respectively be the maximum and the minimum values of f(x) = |1+sin² xcos² x4 sin 4xsin² x1+cos² x4 sin 4xsin² xcos² x1+4 sin 4x|, x ∈ ℝ. Then M⁴ − m⁴ is equal to:

Let A = [a_ij] be a matrix of order 3×3, with a_ij = (√2)^i+j. If the sum of all the elements in the third row of A² is α + β √2, α, β ∈ ℤ, then α + β is equal to:

Let α, β (α ≠ β) be the values of m, for which the equations x + y + z = 1; x + 2y + 4z = m and x + 4y + 10z = m² have infinitely many solutions. Then the value of Σ_n=1¹⁰ (n^α + n^β) is equal to:

If the system of linear equations 3x + y + β z = 3, 2x + α y − z = −3, x + 2y + z = 4 has infinitely many solutions, then the value of 22β − 9α is:

Let a ∈ ℝ and A be a matrix of order 3×3 such that det(A) = −4 and A + I = [1a1210a12], where I is the identity matrix of order 3×3. If det((a + 1) adj((a − 1)A)) is 2^m 3ⁿ, m, n in {0, 1, 2, ..., 20}, then m + n is equal to:

Let A = [α−16β], α > 0, such that det(A) = 0 and α + β = 1. If I denotes 2×2 identity matrix, then the matrix (I + A)⁸ is:

Let A be a 3×3 real matrix such that A²(A − 2I) − 4(A − I) = O, where I and O are the identity and null matrices, respectively. If A⁵ = α A² + β A + γ I, where α, β, and γ are real constants, then α + β + γ is equal to:

Let A be a matrix of order 3×3 and |A| = 5. If |2adj(3A adj(2A))| = 2^α . 3^β . 5^γ, α, β, γ ∈ ℕ, then α + β + γ is equal to.

Let I be the identity matrix of order 3×3 and for the matrix A = [λ234567−12], |A| = −1. Let B be the inverse of the matrix adj(A adj(A²)). Then |(λ B + I)| is equal to ____.

Let A = [cos θ0−sin θ010sin θ0cos θ]. If for some θ ∈ (0, π), A² = A^T, then the sum of the diagonal elements of the matrix (A + I)³ + (A − I)³ − 6A is equal to __________.

Let the matrix A = [100101010] satisfy Aⁿ = Aⁿ⁻² + A² − I for n ≥ 3. Then the sum of all the elements of A⁵⁰ is:

The number of singular matrices of order 2, whose elements are from the set {2,3,6,9}, is ______.

Let the system of equations: 2x + 3y + 5z = 9, 7x + 3y − 2z = 8, 12x + 3y − (4 + λ)z = 16 − μ have infinitely many solutions. Then the radius of the circle centred at (λ, μ) and touching the line 4x = 3y is.

Let A be a 3×3 matrix such that |adj(adj(adj A))| = 81. If S = {n ∈ ℤ : (|adj(adj A)|)^(n−1)2/2 = |A|^3n2−5n−4}, then Σ_{n in S} |Aⁿ²⁺ⁿ| is equal to.

Let A = [22+p2+p+q46+2p8+3p+2q612+3p20+6p+3q]. If det(adj(adj(3A))) = 2^m . 3ⁿ, m, n ∈ ℕ, then m + n is equal to:

For some α, β ∈ ℝ, let A = [α212] and B = [111β] be such that A² − 4A + 2I = B² − 3B + I = O. Then (det(adj(A³ − B³)))² is equal to ____.

If A = [α212], B = [11β1] and A² − 4A + 2I = 0; B² − 2B + I = 0, then |adj(A³ − B³)| is equal to.

If the system of equations 3x + y + 4z = 3, 2x + α y − z = −3, x + 2y + z = 4 has no solution, then the value of α is equal to:

For the matrices A = [3−41−1] and B = [−2949−1318], if (A¹⁵ + B)[xy]^T = [00]^T, then among the following which one is true?

Let A be a 3×3 matrix such that A + A^T = O. If A[1−10]^T = [332]^T, A²[1−10]^T = [−319−24]^T and det(adj(2 adj(A + I))) = (2)^α . (3)^β . (11)^γ, α, β, γ are non−negative integers, then α + β + γ is equal to ____.

If A = [2335], then the determinant of the matrix (A²⁰²⁵ − 3A²⁰²⁴ + A²⁰²³) is.

If X = [xyz]^T is a solution of the system of equations AX = B, where adj A = [422−5051−23] and B = [402]^T, then |x + y + z| is equal to:

Let |A| = 6, where A is a 3×3 matrix. If |adj(3 adj(A² . adj(2A)))| = 2^m . 3ⁿ, m, n ∈ ℕ, then m + n is equal to ____.

Among the statements below: Statement I: If |1cos αcos βcos α1cos γcos βcos γ1| = |0cos αcos βcos α0cos γcos βcos γ0|, then cos² α + cos² β + cos² γ = 3/2. Statement II: If |x²+xx+1x−22x²+3x−13x3x−3x²+2x+32x−12x−1| = px + q, then p² = 196q².

Let A = [02−3−2013−10] and B be a matrix such that B(I − A) = I + A. Then the sum of the diagonal elements of B^T B is equal to ____.

The system of linear equations x + y + z = 6, 2x + 5y + az = 36, x + 2y + 3z = b Has

Let A, B and C be three 2×2 matrices with real entries such that B = (I + A)⁻¹ and A + C = I. If BC = [1−5−12] and CB[x₁x₂]^T = [12−6]^T, then x₁ + x₂ is.

Let A = [3−41−1] and B be two matrices such that A¹⁰⁰ = 100B + I. Then the sum of all the elements of B¹⁰⁰ is ____.

Let A = [121α] and B = [33β2]. If A² − 4A + I = O and B² − 5B − 6I = O, then among the two statements: (S₁): [(B − A)(B + A)]^T = [1315710] and (S₂): det(adj(A + B)) = −5.

Let α, β ∈ ℝ be such that the system of linear equations x + 2y + z = 5, 2x + y + α z = 5, 8x + 4y + β z = 18 has no solution. Then β/α is equal to:

Consider the matrices A = [2−24−2] and B = [3913]. If matrices P and Q are such that PA = B and AQ = B, then the absolute value of the sum of the diagonal elements of 2(P + Q) is ____.

If the system of equations x + 5y + 6z = 4, 2x + 3y + 4z = 7, x + 6y + az = b has infinitely many solutions, then the point (a, b) lies on the line.

Let S = { A = [abcd] : a, b, c, d in {0,1,2,3,4} and A² − 4A + 3I = 0 } be a set of 2×2 matrices. Then the number of matrices in S, for which the sum of the diagonal elements is equal to 4, is:

If the system of equations: x + y + z = 5; x + 2y + 3z = 9; x + 3y + λ z = μ has infinitely many solutions, then the value of λ + μ is:

Consider the system of linear equations in x, y, z: x + 2y + tz = 0; 6x + y + 5tz = 0; 3x + t² y + f(t)z = 0, where f : R → R is a differentiable function. If this system has infinitely many solutions for all t ∈ ℝ, then f

Let A be a 3×3 matrix such that A^T[101]^T = [522]^T, A^T[001]^T = [311]^T, A[101]^T = [344]^T and A[001]^T = [131]^T. If det(A) = 1, then det(adj(A² + A)) is equal to.

Let M be a 3×3 matrix such that M(1,0,0)^T = (1,2,3)^T, M(0,1,0)^T = (0,1,2)^T and M(0,0,1)^T = (−1,1,1)^T. If M(x,y,z)^T = (1,7,11)^T, then x + y + z equals:

If f : N → Z is defined by f(n) = |n−1−5−2n²3(2k+1)2k+1−3n³3k(2k+1)3k(k+2)+1|, k ∈ ℕ, and Σ_n=1^k f(n) = 98, then k is equal to:

Let A = [−11−1101001] satisfy A² + α(adj(adj(A))) + β(adj(A)(adj(adj(A)))) = [2−22−20−100−1] for some α, β ∈ ℤ. Then (α − β)² is equal to ____.

Let A = [100310931] and B = [b_ij], 1 ≤ i, j ≤ 3. If B = A⁹⁹ − I, then the value of (b₃₁ − b₂₁)/b₃₂ is:

The sum of all possible values of θ ∈ [0, 2π], for which the system of equations: x cos 3θ − 8y − 12z = 0; x cos 2θ + 3y + 3z = 0; x + y + 3z = 0 has a non-trivial solution, is equal to:

Let A = [α12230045] and B = [1000−5α004α−2α] + adj(A). If det(B) = 66, then det(adj(A)) equals:

If the system of linear equations: x + y + z = 6, x + 2y + 5z = 10, 2x + 3y + λ z = μ has infinitely many solutions, then the value of λ + μ equals: