Matrices and Determinants are essential topics in JEE Mathematics that test conceptual understanding and application in solving systems of equations, transformations, and inverse operations. This practice set is designed to reinforce these foundational concepts. Topic: Matrices and Determinants 1. If A = [[2, 3], [4, 5]], then |A| is: a) 2 b) -2 c) -1 d) 10 2. If A is a 2×2 matrix such that A² = I, then A is: a) Invertible b) Non-invertible c) Null matrix d) Diagonal 3. For a 3×3 matrix A, the value of |kA| is: a) k³|A| b) k²|A| c) k|A| d) |A| 4. If A is a singular matrix, then |A| is: a) 1 b) -1 c) 0 d) Infinite 5. If A and B are square matrices of the same order and AB = BA, then: a) A and B are equal b) A is symmetric c) A and B commute d) B is a zero matrix 6. If A is an orthogonal matrix, then A⁻¹ is: a) A b) Aᵀ c) -A d) None of these 7. The adjoint of a diagonal matrix is: a) Same diagonal matrix b) Zero matrix c) Inverse matrix d) Transpose 8. The determinant of a skew-symmetric matrix of odd order is: a) 0 b) 1 c) -1 d) Not defined 9. Which matrix has no inverse? a) Identity matrix b) Orthogonal matrix c) Singular matrix d) Diagonal matrix 10. Which of the following is true for all invertible matrices A and B? a) (AB)⁻¹ = A⁻¹B⁻¹ b) (AB)⁻¹ = B⁻¹A⁻¹ c) (A + B)⁻¹ = A⁻¹ + B⁻¹ d) A⁻¹ + B⁻¹ = (AB)⁻¹ Submit Answers Matrices and Determinants PYQs Theory Explanation by Rahul Sir 1. If A = [[2, 3], [4, 5]], then |A| is: Correct Answer: b) -2 Theory: For a 2 × 2 matrix A = [[a, b], [c, d]], its determinant is calculated using |A| = ad − bc. Here, A = [[2, 3], [4, 5]], so |A| = (2)(5) − (3)(4) = 10 − 12 = −2. The determinant is a scalar value associated with a square matrix and is extremely important for studying invertibility, solving linear equations, and calculating areas or volumes. If the determinant is zero, the matrix is singular and has no inverse. A non-zero determinant indicates that the matrix is non-singular and therefore invertible. 2. If A is a 2×2 matrix such that A² = I, then A is: Correct Answer: a) Invertible Theory: Given A² = I, we can write AA = I. By definition, a matrix A is invertible if there exists a matrix B such that AB = BA = I. Here, A itself acts as its inverse because AA = I. Therefore, A⁻¹ = A. This also shows that A must be non-singular, so its determinant cannot be zero. Such a matrix is called an involutory matrix because its square is the identity matrix. Examples include I and certain reflection matrices. Whenever a square matrix satisfies A² = I, its inverse exists immediately without needing the usual adjoint or determinant formula. 3. For a 3×3 matrix A, the value of |kA| is: Correct Answer: a) k³|A| Theory: For an n × n matrix A, multiplying every element of A by a scalar k multiplies its determinant by kⁿ. Therefore, |kA| = kⁿ|A|. Since A is a 3 × 3 matrix, n = 3, giving |kA| = k³|A|. This rule follows from the fact that a determinant is multilinear in the rows or columns. Each of the three rows receives a factor k, so the total factor becomes k × k × k = k³. Students should remember the general formula |kA| = kⁿ|A| for an n-order square matrix. 4. If A is a singular matrix, then |A| is: Correct Answer: c) 0 Theory: A square matrix is called singular precisely when its determinant is zero. Thus, if A is singular, |A| = 0. The determinant provides an important test for invertibility: a square matrix A has an inverse if and only if |A| ≠ 0. When |A| = 0, the rows or columns of the matrix are linearly dependent, meaning one row or column can be expressed as a combination of others. Consequently, the matrix does not have a unique inverse. This concept is fundamental in solving simultaneous linear equations, finding rank, and applying determinant-based methods. 5. If A and B are square matrices of the same order and AB = BA, then: Correct Answer: c) A and B commute Theory: Matrix multiplication is generally not commutative, meaning AB is usually different from BA. However, when two square matrices satisfy AB = BA, they are said to commute with each other. Therefore, the given condition directly means that A and B commute. This does not imply that A = B, that either matrix is symmetric, or that B is a zero matrix. The commutative property is a special relationship between particular matrices. For example, a matrix always commutes with the identity matrix and with its own powers. Recognizing the phrase AB = BA is an important examination shortcut. 6. If A is an orthogonal matrix, then A⁻¹ is: Correct Answer: b) Aᵀ Theory: A square matrix A is orthogonal if AᵀA = AAᵀ = I. Comparing this with the definition of an inverse, the matrix that multiplies A to produce I is its inverse. Hence, for an orthogonal matrix, A⁻¹ = Aᵀ. This is one of the most important properties of orthogonal matrices. It also means that the rows and columns of A form orthonormal sets. Orthogonal matrices preserve lengths and angles under transformations, which is why they are important in geometry, coordinate transformations, and linear algebra. Their determinant is always either +1 or −1. 7. The adjoint of a diagonal matrix is: Correct Answer: a) Same diagonal matrix (only in the general structural sense: it remains diagonal) Theory: The adjoint, or adjugate, of a square matrix is the transpose of its cofactor matrix. For a diagonal matrix, all off-diagonal
