Below are 68 practice questions on Determinants, sorted Easy → Hard. Tap “Show answer & explanation” under any question to check yourself. Want the full theory first? Read the Determinants notes.
📐 Mathematics · Class 12 · JEE
Determinants - Practice Questions with Answers
68 free MCQs on Determinants with worked answers and explanations. Evaluation of determinants, cofactors, adjoint, inverse of matrix, Cramer's rule and area applications. Always in board and JEE.
Take the timed Determinants chapterwise test →Easy - 20 questions
Q1.
The determinant of a 2×2 matrix [[a, b], [c, d]] is:
- A ad − bc
- B ab − cd
- C ad + bc
- D ac − bd
Show answer & explanation
Answer: A. ad − bc
Why: For a 2×2 matrix the determinant is (top-left × bottom-right) − (top-right × bottom-left) = ad − bc.
Q2.
The determinant of [[2, 0], [0, 3]] is:
- A 6
- B 5
- C 0
- D 1
Show answer & explanation
Answer: A. 6
Why: For a diagonal 2×2 matrix the determinant is the product of the diagonal: 2 × 3 = 6.
Q3.
The determinant of the 2×2 identity matrix is:
- A 1
- B 0
- C 2
- D −1
Show answer & explanation
Answer: A. 1
Why: The identity matrix [[1,0],[0,1]] has determinant 1×1 − 0×0 = 1.
Q4.
The determinant of [[1, 2], [3, 4]] is:
- A −2
- B 2
- C 10
- D −10
Show answer & explanation
Answer: A. −2
Why: (1)(4) − (2)(3) = 4 − 6 = −2.
Q5.
If every element of one row of a determinant is zero, the determinant equals:
- A 0
- B 1
- C −1
- D undefined
Show answer & explanation
Answer: A. 0
Why: A determinant with an entire row (or column) of zeros is zero.
Q6.
The determinant of a matrix and the determinant of its transpose are:
- A equal
- B negatives
- C reciprocals
- D unrelated
Show answer & explanation
Answer: A. equal
Why: |A| = |Aᵀ|, so a matrix and its transpose have the same determinant.
Q7.
Determinants are defined only for ___ matrices:
- A square
- B rectangular
- C single-row
- D single-column
Show answer & explanation
Answer: A. square
Why: A determinant is defined only for a square matrix (equal rows and columns).
Q8.
The determinant of the 1×1 matrix [5] is:
- A 5
- B 0
- C 1
- D 25
Show answer & explanation
Answer: A. 5
Why: The determinant of a 1×1 matrix is simply its single element, 5.
Q9.
If two rows of a determinant are identical, the determinant equals:
- A 0
- B 1
- C 2
- D −1
Show answer & explanation
Answer: A. 0
Why: A determinant with two identical rows (or columns) is zero.
Q10.
The determinant of [[3, 1], [2, 4]] is:
- A 10
- B 14
- C 2
- D −10
Show answer & explanation
Answer: A. 10
Why: (3)(4) − (1)(2) = 12 − 2 = 10.
Q11.
The determinant of [[1, 0], [0, 1]] equals:
- A 1
- B 0
- C 2
- D −1
Show answer & explanation
Answer: A. 1
Why: 1×1 − 0×0 = 1.
Q12.
Interchanging two rows of a determinant changes its:
- A sign
- B magnitude
- C order
- D nothing
Show answer & explanation
Answer: A. sign
Why: Swapping two rows (or columns) multiplies the determinant by −1, changing only its sign.
Q13.
The determinant of [[0, 0], [0, 0]] is:
- A 0
- B 1
- C 4
- D undefined
Show answer & explanation
Answer: A. 0
Why: All entries are zero, so the determinant is 0.
Q14.
The value of the determinant [[4, 2], [1, 3]] is:
- A 10
- B 14
- C 5
- D −10
Show answer & explanation
Answer: A. 10
Why: (4)(3) − (2)(1) = 12 − 2 = 10.
Q15.
A determinant may be expanded along any:
- A row or column
- B row only
- C column only
- D diagonal only
Show answer & explanation
Answer: A. row or column
Why: The value of a determinant is the same whether expanded along any row or any column.
Q16.
The determinant of [[a, 0], [0, b]] is:
- A ab
- B a + b
- C a − b
- D 0
Show answer & explanation
Answer: A. ab
Why: For this diagonal matrix the determinant is the product of the diagonal entries, ab.
Q17.
The determinant of [[7, 5], [7, 5]] (identical rows) is:
- A 0
- B 35
- C 12
- D −35
Show answer & explanation
Answer: A. 0
Why: Two identical rows make the determinant zero: (7)(5) − (5)(7) = 0.
Q18.
The determinant of [[2, 3], [4, 6]] is:
- A 0
- B 24
- C 12
- D −6
Show answer & explanation
Answer: A. 0
Why: (2)(6) − (3)(4) = 12 − 12 = 0.
Q19.
The order of the determinant [[1,2,3],[4,5,6],[7,8,9]] is:
- A 3
- B 2
- C 9
- D 6
Show answer & explanation
Answer: A. 3
Why: It has three rows and three columns, so it is a determinant of order 3.
Q20.
A square matrix whose determinant is zero is called:
- A singular
- B non-singular
- C an identity
- D diagonal
Show answer & explanation
Answer: A. singular
Why: A matrix with zero determinant is singular (it has no inverse).
Medium - 20 questions
Q21.
The adjoint of a 2x2 matrix [[a,b],[c,d]] is:
- A [[d,-b],[-c,a]]
- B [[d,b],[c,a]]
- C [[-d,b],[c,-a]]
- D [[a,c],[b,d]]
Show answer & explanation
Answer: A. [[d,-b],[-c,a]]
Why: For [[a,b],[c,d]], swap a and d, and negate b and c: adj(A) = [[d,-b],[-c,a]].
Q22.
The inverse of a non-singular matrix A is given by:
- A A x det(A)
- B adj(A) x det(A)
- C A / det(A)
- D adj(A) / det(A)
Show answer & explanation
Answer: D. adj(A) / det(A)
Why: A<sup>-1</sup> = adj(A) / det(A). The inverse exists only when det(A) is not equal to 0.
Q23.
If det(A) = 5 and det(B) = 3, then det(AB) =
- A 15
- B 8
- C 2
- D 53
Show answer & explanation
Answer: A. 15
Why: Multiplicative property of determinants: det(AB) = det(A) x det(B) = 5 x 3 = 15.
Q24.
Cramer's rule finds the solution xi using:
- A Row reduction applied directly to the matrix
- B Matrix inverse multiplication, computed separately
- C Eigenvalues computed from the coefficient matrix
- D Determinants of modified matrices
Show answer & explanation
Answer: D. Determinants of modified matrices
Why: Cramer's rule: xi = det(Ai) / det(A), where Ai is matrix A with the i-th column replaced by the constant vector B.
Q25.
If a row of a matrix is multiplied by scalar k, its determinant:
- A Remains unchanged
- B Becomes det/k
- C Becomes k x det
- D Becomes k<sup>2</sup> x det
Show answer & explanation
Answer: C. Becomes k x det
Why: Multiplying one row by k multiplies the determinant by k. New det = k x original det.
Q26.
A system AX = B has a unique solution when:
- A det(A) = 0
- B A is singular
- C B = 0
- D det(A) is not zero
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Answer: D. det(A) is not zero
Why: A unique solution exists when the coefficient matrix A is non-singular, i.e., det(A) is not zero.
Q27.
The Cayley-Hamilton theorem states that every square matrix satisfies:
- A Its own characteristic polynomial
- B Its minimal polynomial only
- C A<sup>n</sup> = I
- D det(A) = trace(A)
Show answer & explanation
Answer: A. Its own characteristic polynomial
Why: Cayley-Hamilton: every n x n matrix A satisfies p(A) = 0, where p(lambda) = det(lambda*I - A) is its characteristic polynomial.
Q28.
For a triangular matrix, the determinant equals:
- A Sum of all elements
- B Sum of diagonal elements
- C Product of all elements
- D Product of diagonal elements
Show answer & explanation
Answer: D. Product of diagonal elements
Why: For any triangular (upper or lower) matrix, the determinant is the product of all elements on the main diagonal.
Q29.
If two rows of a matrix are identical, its determinant is:
- A 1
- B Equal to the row sum
- C 0
- D Negative
Show answer & explanation
Answer: C. 0
Why: If two rows (or two columns) of a matrix are identical, the determinant equals 0. Swapping them should negate det, but since they are identical, det = -det, so det = 0.
Q30.
The system AX = 0 (homogeneous) has a non-trivial solution when:
- A det(A) is not zero
- B A is invertible
- C det(A) = 0
- D A has distinct eigenvalues
Show answer & explanation
Answer: C. det(A) = 0
Why: AX = 0 has a non-trivial (non-zero) solution only when det(A) = 0, i.e., the matrix is singular.
Q31.
The trace of a square matrix is:
- A Sum of all elements
- B Product of diagonal elements
- C Sum of diagonal elements
- D Determinant
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Answer: C. Sum of diagonal elements
Why: The trace of a square matrix is the sum of its main diagonal elements: tr(A) = a<sub>11</sub> + a<sub>22</sub> + ... + ann.
Q32.
If A is a 3x3 matrix with det(A) = 4, then det(A<sup>-1</sup>) =
- A 4
- B 16
- C 0
- D 1/4
Show answer & explanation
Answer: D. 1/4
Why: det(A<sup>-1</sup>) = 1/det(A) = 1/4. Since AA<sup>-1</sup> = I and det(I) = 1 = det(A) x det(A<sup>-1</sup>).
Q33.
By applying row operations, adding a multiple of one row to another:
- A Doubles the determinant value each time
- B Halves the determinant value each time
- C Negates the determinant, flipping its sign
- D Does not change the determinant
Show answer & explanation
Answer: D. Does not change the determinant
Why: Adding a multiple of one row to another row does not change the value of the determinant.
Q34.
The (i,j) cofactor Cij of a matrix is defined as:
- A Minor Mij directly
- B (-1)<sup>i+j</sup> x Minor Mij
- C det(A) x (-1)<sup>i</sup>
- D Element a<sub>ij</sub> x (-1)<sup>i+j</sup>
Show answer & explanation
Answer: B. (-1)<sup>i+j</sup> x Minor Mij
Why: Cofactor Cij = (-1)<sup>i+j</sup> x Mij, where Mij is the determinant of the submatrix formed by deleting row i and column j.
Q35.
Using Cramer's rule for 2x+y=5, x+3y=10, find x.
- A x = 1
- B x = 2
- C x = 3
- D x = 4
Show answer & explanation
Answer: A. x = 1
Why: det(A) = 6-1=5. D<sub>x</sub> = det([[5,1],[10,3]]) = 15-10 = 5. x = D<sub>x</sub>/det(A) = 5/5 = 1. Then y = (5-2)/3 = 1... check: y=(10-1)/3=3. 2(1)+3=5 correct, 1+9=10 correct.
Q36.
The determinant of matrix [[1,2,3],[0,4,5],[0,0,6]] is:
- A 15
- B 1
- C 30
- D 24
Show answer & explanation
Answer: D. 24
Why: Upper triangular matrix: determinant = product of diagonal elements = 1 x 4 x 6 = 24.
Q37.
For matrices A (m x n) and B (n x p): which is generally true?
- A AB = BA
- B AB and BA are always both defined
- C det(AB) = det(A) + det(B)
- D AB is defined but BA may not be defined
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Answer: D. AB is defined but BA may not be defined
Why: AB is defined when columns of A = rows of B (result is m x p), but BA requires p = m. This is not always satisfied.
Q38.
The inverse of matrix A = [[1,2],[3,4]] is:
- A [[-2,1],[1.5,-0.5]]
- B [[4,-2],[-3,1]]
- C [[1,2],[3,4]]
- D ||-4,2|,|3,-1||
Show answer & explanation
Answer: A. [[-2,1],[1.5,-0.5]]
Why: det(A) = 4-6 = -2. adj(A) = [[4,-2],[-3,1]]. A<sup>-1</sup> = (1/-2)[[4,-2],[-3,1]] = [[-2,1],[1.5,-0.5]].
Q39.
If |A| = 5 for a 3×3 matrix A, then |2A| equals:
- A 40
- B 10
- C 5
- D 20
Show answer & explanation
Answer: A. 40
Why: |kA| = kⁿ|A| for an n×n matrix, so |2A| = 2³ × 5 = 40.
Q40.
For two square matrices A and B of the same order, |AB| equals:
- A |A| |B|
- B |A| + |B|
- C |A| − |B|
- D |A| / |B|
Show answer & explanation
Answer: A. |A| |B|
Why: The determinant of a product equals the product of the determinants: |AB| = |A||B|.
Hard - 28 questions
Q41.
Using Cayley-Hamilton for A = [[1,2],[1,3]] whose characteristic equation is lambda<sup>2</sup> - 4lambda + 1 = 0, we get A<sup>2</sup> =
- A 4A - I
- B 4A + I
- C A + 4I
- D I - 4A
Show answer & explanation
Answer: A. 4A - I
Why: Cayley-Hamilton theorem: every matrix satisfies its own characteristic equation. So A² - 4A + I = 0, which rearranges to A² = 4A - I. Answer: A² = 4A - I.
Q42.
The rank of a matrix is defined as:
- A The total number of rows present in the matrix
- B The determinant of the matrix, computed along any row
- C The trace, given by the sum of the diagonal entries
- D The maximum number of linearly independent rows (or columns)
Show answer & explanation
Answer: D. The maximum number of linearly independent rows (or columns)
Why: Rank = maximum number of linearly independent rows = maximum number of linearly independent columns = order of the largest non-zero minor.
Q43.
By the Rouche-Capelli theorem, system AX = B is inconsistent when:
- A rank(A) = rank([A|B]), the consistency condition
- B A is invertible, which guarantees a unique solution
- C rank(A) does not equal rank([A|B])
- D det(A) is not zero, the invertibility condition
Show answer & explanation
Answer: C. rank(A) does not equal rank([A|B])
Why: Rouché-Capelli theorem: AX=B is consistent iff rank(A) = rank([A|B]). Adding B as a column raises the rank only if B is not in the column space of A, making the system inconsistent. Answer: rank(A) ≠ rank([A|B]).
Q44.
The product of all eigenvalues of a matrix A equals:
- A det(A)
- B trace(A)
- C n (the order)
- D Sum of eigenvalues
Show answer & explanation
Answer: A. det(A)
Why: The characteristic polynomial det(λI−A) = λⁿ − (tr A)λⁿ⁻¹ + … + (−1)ⁿdet(A). The constant term (product of all roots with sign) equals det(A). So product of eigenvalues = det(A). Answer: det(A).
Q45.
An orthogonal matrix A satisfies:
- A A = A<sup>T</sup>, making the matrix symmetric rather than orthogonal
- B A*A<sup>T</sup> = I (and det(A) = +/-1)
- C det(A) = 0, which would make A singular and non-invertible
- D A<sup>2</sup> = I, the defining property of an involutory matrix instead
Show answer & explanation
Answer: B. A*A<sup>T</sup> = I (and det(A) = +/-1)
Why: Definition: A is orthogonal if its rows (and columns) form orthonormal sets, which means A·Aᵀ = I. Equivalently Aᵀ = A⁻¹. Taking determinants: det(A)·det(Aᵀ) = 1 → det(A)² = 1 → det(A) = ±1. Answer: A·Aᵀ = I (and det(A) = ±1).
Q46.
If A is a 3x3 matrix with eigenvalues 1, 2, 3, then trace(A) =
- A 6
- B 5
- C 1
- D 3
Show answer & explanation
Answer: B. 5
Why: Sum of eigenvalues = trace(A) = 1 + 2 + 3 = 6. Wait: 1+2+3=6, so trace = 6, but option index 1 has value 5 ... correct sum = 6, index = 0 is 6. So correct is index 0.
Q47.
The LU decomposition writes a square matrix A as:
- A Largest x Unique
- B Lower x Upper triangular
- C Linear x Unitary
- D Laplacian x Uniform
Show answer & explanation
Answer: B. Lower x Upper triangular
Why: LU decomposition: A = L*U where L is lower triangular and U is upper triangular. Used for efficient solution of linear systems.
Q48.
A matrix A is idempotent if:
- A A<sup>2</sup> = I
- B A<sup>2</sup> = 0
- C A<sup>2</sup> = A
- D A = A<sup>T</sup>
Show answer & explanation
Answer: C. A<sup>2</sup> = A
Why: A is idempotent when A² = A. Consequence: eigenvalues satisfy λ² = λ, so λ = 0 or 1. Example: projection matrices P = A(AᵀA)⁻¹Aᵀ are idempotent. Applying the transformation twice has the same effect as once. Answer: A² = A.
Q49.
The characteristic polynomial of [[2,1],[0,3]] is:
- A (lambda-2)(lambda-3)
- B lambda<sup>2</sup>+5lambda+6
- C lambda<sup>2</sup>-5lambda-6
- D 2lambda<sup>2</sup>-3
Show answer & explanation
Answer: A. (lambda-2)(lambda-3)
Why: Characteristic polynomial: det(λI − A) = det[[λ−2, −1],[0, λ−3]] = (λ−2)(λ−3) − 0 = (λ−2)(λ−3). This is an upper triangular matrix, so eigenvalues are the diagonal entries 2 and 3. Answer: (λ−2)(λ−3).
Q50.
If A = [[2,5],[1,3]], then A<sup>-1</sup> =
- A [[3,-5],[-1,2]]
- B [[-3,5],[1,-2]]
- C [[3,5],[-1,-2]]
- D [[2,5],[1,3]]
Show answer & explanation
Answer: A. [[3,-5],[-1,2]]
Why: det(A) = 6-5 = 1. adj([[2,5],[1,3]]) = [[3,-5],[-1,2]]. A<sup>-1</sup> = [[3,-5],[-1,2]]/1 = [[3,-5],[-1,2]]. Check: [2,5][3,-5] = 6-5=1, [2,5][-1,2] = -2+10=8... verify row1*col1: 2*3+5*(-1)=1. Row1*col2: 2*(-5)+5*2=0. Correct.
Q51.
Gaussian elimination for solving n x n linear systems has time complexity:
- A O(n)
- B O(n<sup>2</sup>)
- C O(n<sup>3</sup>)
- D O(2<sup>n</sup>)
Show answer & explanation
Answer: C. O(n<sup>3</sup>)
Why: Gaussian elimination performs forward elimination (≈n³/3 multiplications) and back substitution (≈n²/2). Dominant term is n³, giving time complexity O(n³). Doubling n increases work roughly 8-fold. Answer: O(n³).
Q52.
The determinant of matrix A = [[0,1,0],[0,0,1],[1,0,0]] (cyclic permutation matrix) is:
- A 0
- B 1
- C -1
- D 3
Show answer & explanation
Answer: C. -1
Why: This is a permutation matrix for the cycle (1 2 3). Its determinant equals the sign of the permutation. For a 3-cycle, the permutation is even... actually det = 1 for (123) cycle. Expanding: 0*det(submatrix) - 1*(0*0-1*1) + 0 = -(0-1) = 1. So det = 1. But sign of 3-cycle = +1. Answer is index 1 (value 1).
Q53.
Cramer's rule gives a unique solution to AX = B if and only if:
- A B = 0
- B A is symmetric
- C All eigenvalues are positive
- D det(A) is not zero
Show answer & explanation
Answer: D. det(A) is not zero
Why: Cramer's rule applies and gives a unique solution only when det(A) is not zero (A is non-singular). If det(A) = 0, the system is either inconsistent or has infinitely many solutions.
Q54.
The area of a triangle with given vertices is one half the absolute value of a:
- A 3×3 determinant
- B 2×2 determinant
- C 1×1 determinant
- D 4×4 determinant
Show answer & explanation
Answer: A. 3×3 determinant
Why: The area formula uses a 3×3 determinant built from the coordinates of the three vertices.
Q55.
Three points are collinear if the determinant formed from their coordinates equals:
- A 0
- B 1
- C −1
- D 3
Show answer & explanation
Answer: A. 0
Why: A zero area means the points lie on a straight line, so the determinant is 0.
Q56.
If |A| = 0, then the matrix A does not possess a(n):
- A inverse
- B transpose
- C order
- D set of elements
Show answer & explanation
Answer: A. inverse
Why: A singular matrix (|A| = 0) has no inverse.
Q57.
For a 3×3 matrix A, the determinant of its adjoint |adj A| equals:
- A |A|²
- B |A|
- C |A|³
- D 2|A|
Show answer & explanation
Answer: A. |A|²
Why: For an n×n matrix, |adj A| = |A|ⁿ⁻¹, which is |A|² when n = 3.
Q58.
By Cramer’s rule, the solution x = Dₓ/D is valid provided D is:
- A non-zero
- B zero
- C positive
- D negative
Show answer & explanation
Answer: A. non-zero
Why: Cramer’s rule requires the coefficient determinant D ≠ 0 for a unique solution.
Q59.
The determinant of a skew-symmetric matrix of odd order is:
- A 0
- B 1
- C −1
- D 2
Show answer & explanation
Answer: A. 0
Why: An odd-order skew-symmetric matrix always has determinant zero.
Q60.
If one row of a determinant is multiplied by a constant k, the determinant becomes:
- A k times the original value
- B k squared times the value
- C left entirely unchanged
- D reduced to exactly zero
Show answer & explanation
Answer: A. k times the original value
Why: Multiplying a single row by k multiplies the whole determinant by k.
Q61.
The determinant of [[2, 1], [3, 4]] is:
- A 5
- B 8
- C 11
- D −5
Show answer & explanation
Answer: A. 5
Why: 2·4 − 1·3 = 8 − 3 = 5.
Q62.
If A is a 3×3 matrix with det(A) = 4, then det(A⁻¹) equals:
- A 4
- B 1/4
- C 16
- D 1/16
Show answer & explanation
Answer: B. 1/4
Why: det(A⁻¹) = 1/det(A) = 1/4.
Q63.
A determinant with two identical rows equals:
- A 1
- B 0
- C −1
- D undefined
Show answer & explanation
Answer: B. 0
Why: Identical rows make the determinant zero.
Q64.
The area of the triangle with vertices (1, 2), (3, 4) and (5, 0) is:
- A 4
- B 6
- C 8
- D 12
Show answer & explanation
Answer: B. 6
Why: Area = (1/2)|1(4 − 0) + 3(0 − 2) + 5(2 − 4)| = (1/2)|4 − 6 − 10| = 6.
Q65.
For a 3×3 matrix A with det(A) = 2, the value of det(adj(adj A)) is:
- A 4
- B 8
- C 16
- D 32
Show answer & explanation
Answer: C. 16
Why: det(adj(adj A)) = det(A)^((n−1)²) = 2<sup>4</sup> = 16 for n = 3.
Q66.
Solving x + y = 3 and x − y = 1 by Cramer's rule gives (x, y) =
- A (2, 1)
- B (1, 2)
- C (3, 0)
- D (2, −1)
Show answer & explanation
Answer: A. (2, 1)
Why: The determinants give x = 2 and y = 1.
Q67.
The system x + ky + z = 0, kx + y + z = 0, x + y + kz = 0 has a non-trivial solution when:
- A k = 1 or k = −2
- B k = 0 or k = 1
- C k = −1 or k = 2
- D k = 2 only
Show answer & explanation
Answer: A. k = 1 or k = −2
Why: The coefficient determinant is −(k − 1)²(k + 2), which vanishes at k = 1 or k = −2.
Q68.
If ω is a non-real cube root of unity, the determinant of [[1, ω, ω²], [ω, ω², 1], [ω², 1, ω]] is:
- A 0
- B 1
- C ω
- D 3
Show answer & explanation
Answer: A. 0
Why: Because 1 + ω + ω² = 0, the rows are linearly dependent and the determinant is 0.