Take timed practice tests on Matrices and Determinants for JEE Main and JEE Advanced with session-wise drills, score review, and explanation-led revision.
Six 20-question timed sessions plus one 60-question module test. Each question is original and calibrated from the uploaded material pattern without copying PDF wording.
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1. A matrix with m rows and n columns is called:
Explanation: A matrix with m rows and n columns is called an m × n matrix. Order is always stated as rows × columns.
2. If A = [[1, 2], [3, 4]], then Aᵀ is:
Explanation: Transpose swaps rows and columns: (Aᵀ)ᵢⱼ = Aⱼᵢ. Row 1 of A becomes column 1 of Aᵀ: [1,2] → column [1,2]ᵀ.
3. The determinant of [[3, 1], [2, 4]] is:
Explanation: det = 3×4 − 1×2 = 12 − 2 = 10.
4. If A is any square matrix, then AI equals:
Explanation: The identity matrix I is the multiplicative identity: AI = IA = A for any compatible matrix A.
5. A matrix A is symmetric if:
Explanation: A symmetric matrix satisfies Aᵀ = A, meaning the matrix equals its own transpose. Equivalently, aᵢⱼ = aⱼᵢ for all i, j.
6. Two matrices A and B can be added if:
Explanation: Matrix addition is defined only when both matrices have the same number of rows and columns (same order).
7. If A is 2×3 and B is 3×4, then AB has order:
Explanation: For AB to be defined, columns of A must equal rows of B (3=3 ✓). The result has rows of A and columns of B: 2×4.
8. If det(A) = 5 and A is 3×3, then det(2A) is:
Explanation: For an n×n matrix, det(kA) = kⁿ det(A). Here n = 3, k = 2: det(2A) = 2³ × 5 = 8 × 5 = 40.
9. A square matrix A is invertible if and only if:
Explanation: A matrix is invertible (non-singular) iff its determinant is non-zero. If det(A) = 0, the matrix is singular and has no inverse.
10. For a skew-symmetric matrix A of odd order, det(A) is:
Explanation: For a skew-symmetric matrix: det(A) = det(Aᵀ) = det(−A) = (−1)ⁿ det(A). If n is odd: det(A) = −det(A) → 2det(A) = 0 → det(A) = 0.
11. If A and B are square matrices of same order, then det(AB) equals:
Explanation: Product rule for determinants: det(AB) = det(A) × det(B). This is a fundamental property.
12. The system 2x + y = 5, x + 3y = 5 has the solution:
Explanation: Using elimination: from equation 2: x = 5 − 3y. Substitute in eq 1: 2(5−3y) + y = 5 → 10 − 6y + y = 5 → 5y = 5 → y = 1. Then x = 5 − 3 = 2.
13. The trace of a 2×2 matrix [[a, b], [c, d]] is:
Explanation: The trace of a matrix is the sum of diagonal elements. For [[a,b],[c,d]], trace = a + d.
14. If A = [[1, 2], [3, 4]], then A² − 5A equals:
Explanation: A² = [[1+6, 2+8],[3+12, 6+16]] = [[7,10],[15,22]]. 5A = [[5,10],[15,20]]. A²−5A = [[2,0],[0,2]] = 2I. Wait: 7−5=2, 10−10=0, 15−15=0, 22−20=2. So A²−5A = 2I. The Cayley-Hamilton theorem: characteristic polynomial of A is λ²−5λ+det(A) = λ²−5λ+(-2) = λ²−5λ−2=0, so A²=5A+2I, i.e., A²−5A = 2I.
15. For the system AX = 0 to have non-trivial solutions, the condition is:
Explanation: The homogeneous system AX = 0 has non-trivial solutions iff A is singular, i.e., det(A) = 0. If det(A) ≠ 0, the only solution is X = 0 (trivial).
16. If A is a 3×3 matrix with det(A) = 6, then det(adj(A)) equals:
Explanation: For an n×n matrix, det(adj(A)) = (det A)^(n−1). Here n = 3: det(adj(A)) = 6² = 36.
17. If A² = A (idempotent), then the eigenvalues of A are:
Explanation: If λ is an eigenvalue of A with eigenvector v: Av = λv → A²v = λ²v. But A² = A → Av = λv → λ²v = λv → λ(λ−1)v = 0. Since v ≠ 0: λ = 0 or λ = 1.
18. The number of elements in a 3×4 matrix is:
Explanation: A 3×4 matrix has 3 rows and 4 columns, giving 3 × 4 = 12 elements.
19. If A is a 3×3 matrix and det(A) = k, then det(3A) is:
Explanation: det(cA) = cⁿ det(A) for an n×n matrix. Here c = 3, n = 3: det(3A) = 3³ × k = 27k.
20. The inverse of [[cos θ, sin θ], [−sin θ, cos θ]] is:
Explanation: This is a rotation matrix R(θ). Its determinant = cos²θ + sin²θ = 1. For a 2×2 matrix [[a,b],[c,d]] with det=1: inverse = [[d,−b],[−c,a]] = [[cos θ, −sin θ],[sin θ, cos θ]] = R(−θ).