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
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 cofactors are zero, so its adjoint is also a diagonal matrix. However, the diagonal entries are not generally unchanged. For example, if A = diag(a, d) = [[a, 0], [0, d]], then adj(A) = [[d, 0], [0, a]]. Thus, option (a) is the intended answer only if “same diagonal matrix” means the same diagonal form, not identical diagonal entries. The key theory is that the adjoint of a diagonal matrix is itself diagonal.
8. The determinant of a skew-symmetric matrix of odd order is:
Correct Answer: a) 0
Theory: A skew-symmetric matrix satisfies Aᵀ = −A. For any square matrix, |Aᵀ| = |A|. Therefore, for a skew-symmetric matrix, |A| = |Aᵀ| = |−A|. For an n × n matrix, |−A| = (−1)ⁿ|A|. If n is odd, (−1)ⁿ = −1, so |A| = −|A|, which gives 2|A| = 0 and hence |A| = 0. Therefore, every skew-symmetric matrix of odd order is singular. This is a standard determinant result and is frequently tested in matrix and determinant problems.
9. Which matrix has no inverse?
Correct Answer: c) Singular matrix
Theory: A square matrix has an inverse if and only if its determinant is non-zero. A singular matrix is defined as a square matrix whose determinant is zero. Therefore, a singular matrix has no inverse. In contrast, the identity matrix is always invertible, an orthogonal matrix has inverse equal to its transpose, and a diagonal matrix is invertible whenever all its diagonal entries are non-zero. For a diagonal matrix, if even one diagonal entry is zero, its determinant becomes zero and the matrix becomes singular. Hence, the decisive test for invertibility is always the determinant.
10. Which of the following is true for all invertible matrices A and B?
Correct Answer: b) (AB)⁻¹ = B⁻¹A⁻¹
Theory: For two invertible matrices A and B, the inverse of their product is obtained by reversing the order of the individual inverses. Thus, (AB)⁻¹ = B⁻¹A⁻¹. To verify this, multiply AB by B⁻¹A⁻¹: (AB)(B⁻¹A⁻¹) = A(BB⁻¹)A⁻¹ = AIA⁻¹ = I. The reverse product also gives I. The order cannot generally be written as A⁻¹B⁻¹ because matrix multiplication is not commutative. This reverse-order rule is a fundamental property of matrix inverses and should be memorized carefully for JEE-level matrix problems.
Important Note: In Question 7, the phrase “same diagonal matrix” can be misleading. The adjoint of a diagonal matrix is diagonal, but its diagonal entries are generally changed. For example, adj(diag(a,d)) = diag(d,a) in the 2×2 case.
