68 practice questions on Determinants, sorted Easy → Hard. Try each one first, then open its answer page for the worked explanation. Want the full theory first? Read the Determinants notes.
📐 Mathematics · Class 12 · JEE
Determinants - Practice Questions with Answers
68 free MCQs on Determinants, each with its own worked answer and explanation. 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
Q5.
If every element of one row of a determinant is zero, the determinant equals:
- A 0
- B 1
- C −1
- D undefined
Q6.
The determinant of a matrix and the determinant of its transpose are:
- A equal
- B negatives
- C reciprocals
- D unrelated
Q7.
Determinants are defined only for ___ matrices:
- A square
- B rectangular
- C single-row
- D single-column
Q9.
If two rows of a determinant are identical, the determinant equals:
- A 0
- B 1
- C 2
- D −1
Q12.
Interchanging two rows of a determinant changes its:
- A sign
- B magnitude
- C order
- D nothing
Q15.
A determinant may be expanded along any:
- A row or column
- B row only
- C column only
- D diagonal only
Q17.
The determinant of [[7, 5], [7, 5]] (identical rows) is:
- A 0
- B 35
- C 12
- D −35
Q19.
The order of the determinant [[1,2,3],[4,5,6],[7,8,9]] is:
- A 3
- B 2
- C 9
- D 6
Q20.
A square matrix whose determinant is zero is called:
- A singular
- B non-singular
- C an identity
- D diagonal
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]]
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)
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
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
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
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)
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
Q29.
If two rows of a matrix are identical, its determinant is:
- A 1
- B Equal to the row sum
- C 0
- D Negative
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
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
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
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
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>
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
Q36.
The determinant of matrix [[1,2,3],[0,4,5],[0,0,6]] is:
- A 15
- B 1
- C 30
- D 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
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||
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|
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
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)
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
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
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
Q46.
If A is a 3x3 matrix with eigenvalues 1, 2, 3, then trace(A) =
- A 6
- B 5
- C 1
- D 3
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
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>
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
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]]
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>)
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
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
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
Q55.
Three points are collinear if the determinant formed from their coordinates equals:
- A 0
- B 1
- C −1
- D 3
Q56.
If |A| = 0, then the matrix A does not possess a(n):
- A inverse
- B transpose
- C order
- D set of elements
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|
Q58.
By Cramer’s rule, the solution x = Dₓ/D is valid provided D is:
- A non-zero
- B zero
- C positive
- D negative
Q59.
The determinant of a skew-symmetric matrix of odd order is:
- A 0
- B 1
- C −1
- D 2
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
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
Q64.
The area of the triangle with vertices (1, 2), (3, 4) and (5, 0) is:
- A 4
- B 6
- C 8
- D 12
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
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)
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
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
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