Determinants MCQs With Answer

Determinants MCQs With Answer provide B. Pharm students a focused way to master matrix determinants, their properties, and practical applications in pharmacokinetics and drug formulation modeling. This concise, Student-friendly post covers key terms like matrix determinant, Cramer’s rule, adjoint matrix, inverse matrix, Laplace expansion, and singular vs. nonsingular matrices. Questions emphasize calculation skills (2×2 and 3×3), theoretical properties (row operations, triangular matrices, multiplicative property), and applied uses for solving linear systems in pharmacology. Clear explanations and correct options help build exam readiness and quantitative confidence. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the determinant of the 2×2 matrix [[2,3],[1,4]]?

• 5
• 11
• 7
• 2

Correct Answer: 5

Q2. If A is a 3×3 triangular matrix with diagonal entries 2, -1, and 3, what is det(A)?

• 6
• -6
• -1
• -5

Correct Answer: -6

Q3. Which statement is true about determinant and transpose for any square matrix A?

• det(A^T) = -det(A)
• det(A^T) = det(A)
• det(A^T) = 1/det(A)
• det(A^T) = 0

Correct Answer: det(A^T) = det(A)

Q4. If two rows of a matrix are swapped, how does the determinant change?

• The determinant is squared
• The determinant remains the same
• The determinant changes sign
• The determinant becomes zero

Correct Answer: The determinant changes sign

Q5. Which of the following indicates that a square matrix is singular?

• determinant is nonzero
• determinant is zero
• matrix is diagonal
• matrix is symmetric

Correct Answer: determinant is zero

Q6. For a scalar k and n×n matrix A, what is det(kA)?

• k * det(A)
• k^n * det(A)
• det(A)^k
• det(A)/k

Correct Answer: k^n * det(A)

Q7. Determinant of the identity matrix I_n is:

• 0
• n
• 1
• -1

Correct Answer: 1

Q8. If det(A) = 5 and det(B) = -2 for two n×n matrices, what is det(AB)?

• -7
• -10
• 10
• 7

Correct Answer: -10

Q9. Which expansion method uses minors and cofactors to compute a determinant?

• Gaussian elimination
• Laplace expansion
• Fourier transform
• Euler method

Correct Answer: Laplace expansion

Q10. Determinant of a matrix product satisfies which property?

• det(AB) = det(A) + det(B)
• det(AB) = det(A) * det(B)
• det(AB) = det(A – B)
• det(AB) = det(A)/det(B)

Correct Answer: det(AB) = det(A) * det(B)

Q11. For the 2×2 matrix [[a,b],[c,d]], the determinant formula is:

• ad + bc
• ad – bc
• ab – cd
• ac – bd

Correct Answer: ad – bc

Q12. How does adding a multiple of one row to another row affect the determinant?

• Multiplies determinant by that multiple
• Leaves determinant unchanged
• Changes sign of determinant
• Makes determinant zero

Correct Answer: Leaves determinant unchanged

Q13. Determinant of an orthogonal matrix is:

• 0
• 1 or -1
• Always 1
• Always -1

Correct Answer: 1 or -1

Q14. The determinant equals the product of eigenvalues. What does this imply if one eigenvalue is zero?

• All eigenvalues are zero
• The determinant is zero
• The determinant equals product excluding zero
• Matrix is diagonalizable

Correct Answer: The determinant is zero

Q15. Which formula gives the inverse of a 2×2 matrix [[a,b],[c,d]] when det ≠ 0?

• (1/det) * [[d,-b],[-c,a]]
• (1/det) * [[a,b],[c,d]]
• (1/det) * [[-d,b],[c,-a]]
• Transpose of matrix

Correct Answer: (1/det) * [[d,-b],[-c,a]]

Q16. When expanding a determinant along a row, what is the sign factor for element in row i, column j?

• (-1)^(i+j)
• (-1)^(i-j)
• (-1)^(i*j)
• (-1)^(i+j+1)

Correct Answer: (-1)^(i+j)

Q17. The adjugate (adjoint) of a matrix is used to compute:

• determinant only
• inverse through adj(A)/det(A)
• trace of the matrix
• rank of the matrix

Correct Answer: inverse through adj(A)/det(A)

Q18. If a matrix has two identical rows, its determinant is:

• Positive
• Negative
• Zero
• Undefined

Correct Answer: Zero

Q19. Which operation multiplies the determinant by a scalar k when applied to a matrix?

• Multiplying one row by k
• Adding k to one row
• Swapping two rows
• Transposing the matrix

Correct Answer: Multiplying one row by k

Q20. Cramer’s rule solves linear systems using determinants. For two equations in two unknowns, what does Cramer’s rule require?

• determinant of coefficient matrix nonzero
• determinant zero
• coefficients must be integers
• matrix must be symmetric

Correct Answer: determinant of coefficient matrix nonzero

Q21. The determinant of a block diagonal matrix equals:

• sum of determinants of blocks
• product of determinants of diagonal blocks
• determinant of the first block only
• difference of determinants of blocks

Correct Answer: product of determinants of diagonal blocks

Q22. In pharmacokinetics, solving compartmental linear equations often uses:

• Laplace expansion without determinants
• Cramer’s rule or matrix inversion using determinants
• Nonlinear regression only
• Graphical methods exclusively

Correct Answer: Cramer’s rule or matrix inversion using determinants

Q23. For a 3×3 matrix, which method can simplify determinant calculation by introducing zeros?

• Laplace expansion only
• Row operations to get triangular form
• Random shuffling of rows
• Scaling rows to large numbers

Correct Answer: Row operations to get triangular form

Q24. The determinant of a 3×3 matrix equals the scalar triple product of its row vectors. This geometric interpretation measures:

• Area of a parallelogram
• Volume of a parallelepiped
• Length of a vector
• Angle between vectors

Correct Answer: Volume of a parallelepiped

Q25. If det(A) = 4, what is det(A^-1)?

• 4
• 1/4
• -4
• 0

Correct Answer: 1/4

Q26. Which of these is NOT a property of determinants?

• Multiplicative over matrix product
• Change sign when rows swapped
• Linear in each row separately
• Invariant under multiplying all rows by different scalars without change

Correct Answer: Invariant under multiplying all rows by different scalars without change

Q27. For matrix A, det(2A) for a 3×3 matrix with det(A)=3 equals:

• 6
• 8
• 24
• 48

Correct Answer: 24

Q28. What is the minor M_ij of an element a_ij in a matrix?

• Determinant of the matrix after deleting row i and column j
• Cofactor multiplied by (-1)^(i+j)
• Element a_ij squared
• Inverse of element a_ij

Correct Answer: Determinant of the matrix after deleting row i and column j

Q29. The cofactor C_ij is defined as:

• M_ij only
• (-1)^(i+j) * M_ij
• M_ij / (i+j)
• Product of row i elements

Correct Answer: (-1)^(i+j) * M_ij

Q30. If determinant of A is -3, what is determinant of -A for a 2×2 matrix?

• -3
• 3
• 9
• -9

Correct Answer: 3

Q31. Which determinant technique is computationally efficient for large matrices?

• Direct Laplace expansion
• LU decomposition and product of diagonal entries
• Brute-force permutation formula
• Manual cofactor expansion for every element

Correct Answer: LU decomposition and product of diagonal entries

Q32. Determinant of the matrix [[1,2,3],[0,4,5],[0,0,6]] equals:

• 120
• 24
• 6
• 0

Correct Answer: 24

Q33. The Jacobian determinant is used in pharmacology for:

• Solving ordinary differential equations numerically
• Change of variables in multiple integrals
• Estimating half-life directly
• Measuring partition coefficient

Correct Answer: Change of variables in multiple integrals

Q34. If A has two proportional rows, determinant of A is:

• Nonzero
• Zero
• Equal to trace
• Infinite

Correct Answer: Zero

Q35. Which is the determinant of [[0,1],[-1,0]] representing a 90° rotation?

• 0
• 1
• -1
• Undefined

Correct Answer: 1

Q36. When using Gaussian elimination to compute a determinant, which row operation requires scaling the determinant?

• Adding a multiple of one row to another
• Swapping two rows
• Multiplying a row by a scalar
• Transposing the matrix

Correct Answer: Multiplying a row by a scalar

Q37. In the context of linear systems, a zero determinant means:

• Unique solution exists
• No solution or infinitely many solutions
• System is overdetermined only
• System is homogeneous only

Correct Answer: No solution or infinitely many solutions

Q38. The determinant of a 3×3 matrix can be computed by Sarrus’ rule. Sarrus’ rule applies to:

• Only 2×2 matrices
• Only 3×3 matrices
• Any n×n matrix
• Only triangular matrices

Correct Answer: Only 3×3 matrices

Q39. For square matrices A and B of same size, if B is obtained by multiplying a row of A by 3, then det(B) = ?

• det(A) / 3
• det(A) + 3
• 3 * det(A)
• det(A)^3

Correct Answer: 3 * det(A)

Q40. What is an immediate test for linear dependence of rows using determinant?

• If determinant is negative
• If determinant equals trace
• If determinant equals zero
• If determinant equals product of diagonals

Correct Answer: If determinant equals zero

Q41. For a 3×3 matrix with two zero columns, determinant is:

• Dependent on nonzero column
• Zero
• Product of nonzero entries
• Equal to trace

Correct Answer: Zero

Q42. Which expression gives determinant of rotation by angle θ in 2D?

• cosθ + sinθ
• cos^2θ + sin^2θ
• 1
• cos2θ

Correct Answer: 1

Q43. If det(A) = 0, which of the following is true about A?

• A is invertible
• A has full rank
• Columns of A are linearly dependent
• All eigenvalues are nonzero

Correct Answer: Columns of A are linearly dependent

Q44. Which of these is a correct step to compute determinant via LU decomposition?

• det(A) = det(L) + det(U)
• det(A) = det(L) * det(U) and det(L)=1 for Doolittle
• det(A) = trace(L) * trace(U)
• det(A) = det(U) only

Correct Answer: det(A) = det(L) * det(U) and det(L)=1 for Doolittle

Q45. Which determinant value indicates a volume-preserving linear map in 3D?

• 0
• -1
• 1
• Any positive integer

Correct Answer: 1

Q46. Using Cramer’s rule for a 2×2 system, x = det(A_x)/det(A). What does A_x represent?

• Coefficient matrix with first column replaced by constants
• Inverse of A
• Transpose of A
• Matrix of cofactors

Correct Answer: Coefficient matrix with first column replaced by constants

Q47. Which determinant property helps show det(e^A) = e^{trace(A)} for diagonalizable A?

• Multiplicative property and eigenvalue product relation
• Swapping rows changes sign
• Triangular matrix determinant rule only
• Adjugate inverse formula

Correct Answer: Multiplicative property and eigenvalue product relation

Q48. For a 4×4 matrix, which approach reduces computation effort for determinant?

• Direct expansion along first row always
• Use row operations to create zeros then compute product of pivots
• Compute determinant by evaluating all 24 permutations manually
• Compute determinant by squaring entries

Correct Answer: Use row operations to create zeros then compute product of pivots

Q49. The determinant of a matrix can be interpreted as:

• Scale factor of the linear transformation on volume
• Sum of singular values
• Maximum eigenvalue only
• Number of solutions to a system

Correct Answer: Scale factor of the linear transformation on volume

Q50. In applied pharmaceutical modeling, why are determinants important for solving linear algebraic systems?

• They directly give concentration values without equations
• They determine invertibility and enable analytical solution methods like Cramer’s rule and matrix inversion
• They are not used in modeling
• They only apply to nonlinear systems

Correct Answer: They determine invertibility and enable analytical solution methods like Cramer’s rule and matrix inversion

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