Determinants — Class 12 MCQs with Answers

Determinants — important questions & MCQs with answers (Class 12 Mathematics)

14 solved questions from this chapter's difficulty levels, each with its answer and explanation. The other 76 are timed and scored when you sign in.

1. Q1Easy

   det of [[1,2],[3,4]]:

   A.10

   B.2

   C.−2✓ Correct

   D.0

   Answer: C. −2

   Explanation: det = ad − bc = 1×4 − 2×3 = 4 − 6 = −2. Mixing up the diagonal order (bc − ad) gives +2, the tempting wrong-sign distractor.

2. Q2Easy

   Matrix is singular if det =

   A.0✓ Correct

   B.1

   C.−1

   D.any

   Answer: A. 0

   Explanation: A matrix is singular exactly when det(A) = 0, since then adj(A)/det(A) is undefined and no inverse exists — not det = 1 or any other value.

3. Q3Easy

   det([[a, b], [c, d]]) =

   A.ab − cd

   B.ad − bc✓ Correct

   C.ad + bc

   D.cd − ab

   Answer: B. ad − bc

   Explanation: For a 2×2 matrix, det = ad − bc: multiply the main diagonal, subtract the product of the off-diagonal, e.g. det[[1,2],[3,4]] = 4 − 6 = −2.

4. Q4Easy

   3×3 determinant is computed by:

   A.Multiplying diagonal

   B.Adding rows

   C.Expanding along a row or column✓ Correct

   D.Cannot compute

   Answer: C. Expanding along a row or column

   Explanation: A 3×3 determinant is found by cofactor expansion along any row or column — never by multiplying the diagonal or adding rows, which only work for special cases.

5. Q5Easy

   Minor M_ij of element a_ij in matrix A:

   A.Determinant of matrix obtained by deleting row i and column j✓ Correct

   B.Same as a_ij

   C.Sum of column j

   D.Product

   Answer: A. Determinant of matrix obtained by deleting row i and column j

   Explanation: The minor Mᵢⱼ is the determinant of the smaller matrix left after deleting row i and column j — it is a number derived from the matrix, not the entry aᵢⱼ itself.

6. Q6Easy

   Cofactor A_ij = ?

   A.a_ij

   B.M_ij only

   C.(−1)^(i+j) M_ij✓ Correct

   D.Sum

   Answer: C. (−1)^(i+j) M_ij

   Explanation: Cofactor Aᵢⱼ = (−1)^(i+j)·Mᵢⱼ: take the minor and attach a sign based on position, e.g. A₁₂ = −M₁₂. Forgetting the sign gives just the minor, a common slip.

7. Q7Medium

   Swapping two rows changes det by:

   A.Doubles

   B.Sign (multiplied by −1)✓ Correct

   C.Halves

   D.Stays same

   Answer: B. Sign (multiplied by −1)

   Explanation: Swapping any two rows (or columns) multiplies the determinant by −1, e.g. det[[1,2],[3,4]] = −2 becomes +2 after swapping — it never doubles or halves.

8. Q8Medium

   A⁻¹ exists iff:

   A.A is square only

   B.det(A) = 0

   C.det(A) ≠ 0✓ Correct

   D.Always

   Answer: C. det(A) ≠ 0

   Explanation: A⁻¹ = adj(A)/det(A) only makes sense when det(A) ≠ 0; being square is necessary but not sufficient — a square matrix can still have det = 0.

9. Q9Medium

   If the matrix [[x, 2], [3, 4]] is singular, then x is:

   A.3/2✓ Correct

   B.6

   C.2/3

   D.4

   Answer: A. 3/2

   Explanation: Singularity requires 4x − 6 = 0, so x = 3/2.

10. Q10Medium

    det([[1, 0, 0], [0, 2, 0], [0, 0, 3]]) =

    A.3

    B.5

    C.1

    D.6✓ Correct

    Answer: D. 6

    Explanation: A diagonal matrix's determinant is just the product of its diagonal entries: 1×2×3 = 6 — every off-diagonal cofactor term vanishes since those entries are 0.

11. Q11Medium

    If A is 3×3 with det(A) = 5, then det(3A) =

    A.135✓ Correct

    B.45

    C.5

    D.15

    Answer: A. 135

    Explanation: For an n×n matrix, det(kA) = kⁿ·det(A), not k·det(A): here n = 3, so det(3A) = 3³×5 = 27×5 = 135, not 3×5 = 15.

12. Q12Hard

    det(AB) =

    A.det(A) · det(B)✓ Correct

    B.det(A) + det(B)

    C.det(A)/det(B)

    D.det(A) − det(B)

    Answer: A. det(A) · det(B)

    Explanation: det(AB) = det(A)·det(B) always, e.g. det(A)=2, det(B)=3 gives det(AB)=6 — determinants multiply under matrix products, they never add or subtract.

13. Q13Hard

    Cramer's rule applies when:

    A.Never

    B.Det is zero

    C.Always

    D.Coefficient det ≠ 0✓ Correct

    Answer: D. Coefficient det ≠ 0

    Explanation: Cramer's rule needs the coefficient determinant D ≠ 0, since each unknown is Dᵢ/D — a zero D makes that division undefined, not automatically 'no solution'.

14. Q14Hard

    If A = [[2, 3], [1, 4]], then the element in the first row and first column of A⁻¹ is:

    A.1/5

    B.4/5✓ Correct

    C.−3/5

    D.2/5

    Answer: B. 4/5

    Explanation: The determinant is 5 and the adjoint has 4 in that position, so the entry is 4/5.

Determinants — FAQs

More Class 12 Mathematics chapters

• 01Relations and Functions
• 02Inverse Trigonometric Functions
• 03Matrices
• 05Continuity and Differentiability
• 06Application of Derivatives
• 07Integrals

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