Operations on matrices MCQs With Answer

Operations on matrices MCQs With Answer — This concise introduction helps B. Pharm students master matrix operations used in pharmaceutical calculations, formulation modeling, and pharmacokinetics. Learn essential keywords like operations on matrices, matrix addition, matrix multiplication, transpose, determinant, inverse matrix, identity matrix, and rank. Understanding these operations enhances problem-solving for linear systems, drug dosage matrices, and laboratory data analysis. The following MCQs are tailored to B. Pharm curriculum level, mixing theory and applied examples to build confidence and exam readiness. Clear explanations and correct answers will reinforce concepts and calculation techniques. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the result of adding two matrices of sizes 2×3 and 2×3?

• A 2×3 matrix
• A 3×2 matrix
• A 2×2 matrix
• Undefined

Correct Answer: A 2×3 matrix

Q2. Which operation is not defined for a 2×3 matrix?

• Addition with another 2×3 matrix
• Multiplication by a 3×2 matrix
• Transpose
• Multiplication by a 2×3 matrix

Correct Answer: Multiplication by a 2×3 matrix

Q3. If A is a 3×3 identity matrix I3, what is A * I3?

• Zero matrix
• Matrix A
• Transpose of A
• Inverse of A

Correct Answer: Matrix A

Q4. The determinant of a 2×2 matrix [[a,b],[c,d]] is:

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

Correct Answer: ad – bc

Q5. For a square matrix A, A^T denotes:

• Inverse of A
• Transpose of A
• Trace of A
• Adjoint of A

Correct Answer: Transpose of A

Q6. A matrix that has nonzero determinant is called:

• Singular
• Orthogonal
• Invertible (nonsingular)
• Diagonalizable

Correct Answer: Invertible (nonsingular)

Q7. The inverse of a 2×2 matrix [[a,b],[c,d]] (if exists) is:

• 1/(ad-bc) * [[d,-b],[-c,a]]
• 1/(ad+bc) * [[d,b],[c,a]]
• [[d,-b],[-c,a]]
• [[a,d],[b,c]]

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

Q8. If A is m×n and B is n×p, the product AB has size:

• m×n
• n×p
• m×p
• p×m

Correct Answer: m×p

Q9. Matrix multiplication is:

• Commutative for all matrices
• Associative but not generally commutative
• Neither associative nor commutative
• Always commutative for square matrices

Correct Answer: Associative but not generally commutative

Q10. The identity matrix I satisfies which property for any conformable matrix A?

• IA = 0
• AI = A only if A is square
• IA = AI = A
• I = A^-1

Correct Answer: IA = AI = A

Q11. The zero matrix acts as what under matrix addition?

• Multiplicative identity
• Additive identity
• Inverse element
• Determinant provider

Correct Answer: Additive identity

Q12. For a scalar k and matrix A, k(A + B) equals:

• kA + B
• A + kB
• kA + kB
• k(AB)

Correct Answer: kA + kB

Q13. A symmetric matrix satisfies which condition?

• A^T = -A
• A^T = A
• A^2 = I
• Determinant zero

Correct Answer: A^T = A

Q14. A skew-symmetric matrix A satisfies:

• A^T = A
• A^T = -A
• A is invertible
• All diagonal entries are 1

Correct Answer: A^T = -A

Q15. The rank of a matrix is defined as:

• Number of zero rows
• Maximum number of linearly independent rows or columns
• Number of nonzero entries
• Sum of diagonal elements

Correct Answer: Maximum number of linearly independent rows or columns

Q16. Elementary row operations include all except:

• Swapping two rows
• Multiplying a row by a nonzero scalar
• Adding a multiple of one row to another
• Changing the order of columns arbitrarily

Correct Answer: Changing the order of columns arbitrarily

Q17. Which matrix property is preserved by elementary row operations?

• Determinant always unchanged
• Row space dimension (rank)
• Column order
• Exact entries

Correct Answer: Row space dimension (rank)

Q18. The trace of a square matrix is:

• Product of diagonal entries
• Sum of diagonal entries
• Number of rows
• Determinant

Correct Answer: Sum of diagonal entries

Q19. If A and B are invertible matrices of the same size, (AB)^-1 equals:

• A^-1 B^-1
• B^-1 A^-1
• A B^-1
• B A

Correct Answer: B^-1 A^-1

Q20. Which method is commonly used to find the inverse of a matrix by row operations?

• Gaussian elimination on [A | I]
• Computing only determinant
• Transposing and dividing by determinant
• Multiplying by zero matrix

Correct Answer: Gaussian elimination on [A | I]

Q21. A necessary condition for a square matrix to be invertible is:

• Determinant equals zero
• Rank less than size
• Determinant is nonzero
• Matrix is symmetric

Correct Answer: Determinant is nonzero

Q22. The transpose of a product (AB)^T equals:

• A^T B^T
• B^T A^T
• A B^T
• B A^T

Correct Answer: B^T A^T

Q23. In pharmaceutical stoichiometry, matrices help solve linear systems representing:

• Only chemical names
• Concentrations and component balances
• Visual inspection data
• Temperature readings only

Correct Answer: Concentrations and component balances

Q24. The determinant of a triangular (upper or lower) square matrix equals:

• Sum of diagonal entries
• Product of diagonal entries
• Zero always
• Trace squared

Correct Answer: Product of diagonal entries

Q25. If two rows of a square matrix are identical, the determinant is:

• Positive
• Negative
• Zero
• Undefined

Correct Answer: Zero

Q26. Which condition indicates a matrix is orthogonal?

• A^T = A
• A^T A = I
• Determinant zero
• All entries equal

Correct Answer: A^T A = I

Q27. Multiplying a matrix by a scalar affects which property?

• Number of rows and columns
• Linear independence unaffected if scalar nonzero
• Rank always becomes zero
• Transpose changes to inverse

Correct Answer: Linear independence unaffected if scalar nonzero

Q28. The adjugate (adjoint) of a matrix is used to compute:

• Matrix transpose
• Matrix inverse (via adj(A)/det(A))
• Rank directly
• Only eigenvalues

Correct Answer: Matrix inverse (via adj(A)/det(A))

Q29. Which statement about determinants is true?

• Determinant of product equals sum of determinants
• Determinant of product equals product of determinants
• Determinant is invariant under row addition
• Determinant equals trace

Correct Answer: Determinant of product equals product of determinants

Q30. A 3×3 matrix with rank 2 is:

• Invertible
• Singular
• Diagonalizable always
• Orthogonal

Correct Answer: Singular

Q31. Which of the following is true for row-reduced echelon form (RREF)?

• Every pivot column has leading 1 and zeros elsewhere
• Pivots may be any nonzero number
• It is unique only for square matrices
• It increases the determinant

Correct Answer: Every pivot column has leading 1 and zeros elsewhere

Q32. The solution set of a homogeneous system Ax = 0 is:

• Always only the zero vector
• A linear subspace (possibly nontrivial)
• Not a vector space
• Only for invertible A has nontrivial solutions

Correct Answer: A linear subspace (possibly nontrivial)

Q33. Which property holds for scalar multiplication and matrix addition?

• Distributivity: k(A + B) = kA + kB
• k(A + B) = (kA)B
• Scalar multiplication is non-distributive
• A + B = AB for scalars

Correct Answer: Distributivity: k(A + B) = kA + kB

Q34. To solve linear equations in B. Pharm for mixture problems, which matrix operation is frequently used?

• Matrix inversion or Gaussian elimination
• Only matrix transposition
• Computing determinants only
• Creating zero matrices always

Correct Answer: Matrix inversion or Gaussian elimination

Q35. If A is 2×2 with determinant -5, determinant of 3A is:

• -5
• -15
• -45
• -20

Correct Answer: -45

Q36. The row space and column space of a matrix have the same:

• Number of rows
• Number of columns
• Dimension (rank)
• Entries

Correct Answer: Dimension (rank)

Q37. A square matrix A is diagonalizable if it:

• Is already diagonal only
• Has a full set of linearly independent eigenvectors
• Has determinant zero
• Has equal rows

Correct Answer: Has a full set of linearly independent eigenvectors

Q38. Which statement about transpose is correct?

• (A + B)^T = A^T + B^T
• (A + B)^T = A + B
• A^T = A^-1 for all A
• Transpose changes matrix size unpredictably

Correct Answer: (A + B)^T = A^T + B^T

Q39. In pharmacokinetic modeling, matrices can represent:

• Compartments and transfer rates between them
• Only pill shapes
• Color of solutions
• Temperature sensors exclusively

Correct Answer: Compartments and transfer rates between them

Q40. Which of these matrices has determinant equal to 1 and preserves lengths?

• Singular matrix
• Orthogonal matrix with determinant 1 (special orthogonal)
• Diagonal matrix with zeros
• Any symmetric matrix

Correct Answer: Orthogonal matrix with determinant 1 (special orthogonal)

Q41. If A is 4×4 and rank(A) = 4, then A is:

• Singular
• Invertible
• Zero matrix
• Rectangular

Correct Answer: Invertible

Q42. Which matrix equation expresses that vectors v1,…,vn are columns of A?

• A = [v1 v2 … vn]
• A = v1 + v2 + … + vn
• A = determinant(v1,…,vn)
• A = v1 * v2 * … * vn

Correct Answer: A = [v1 v2 … vn]

Q43. The null space (kernel) of A consists of:

• All x such that Ax = 0
• All x such that xA = 0
• Only zero vector always
• All eigenvalues of A

Correct Answer: All x such that Ax = 0

Q44. Which of the following transforms a matrix into row echelon form?

• Adding columns randomly
• Elementary row operations (Gaussian elimination)
• Multiplying by identity only
• Taking transpose repeatedly

Correct Answer: Elementary row operations (Gaussian elimination)

Q45. For matrices A and B, if AB = 0 (zero matrix), then:

• Either A = 0 or B = 0 must hold
• Both A and B are invertible
• At least one of A or B may be singular, but not necessarily zero
• Determinant of A equals determinant of B

Correct Answer: At least one of A or B may be singular, but not necessarily zero

Q46. Which concept helps reduce solving large linear systems in drug formulation?

• Using random matrices
• LU decomposition or Gaussian elimination
• Only computing determinants
• Ignoring linear dependence

Correct Answer: LU decomposition or Gaussian elimination

Q47. A square matrix with zeros everywhere except possible diagonal entries is called:

• Zero matrix
• Diagonal matrix
• Symmetric matrix only
• Singular matrix always

Correct Answer: Diagonal matrix

Q48. The process of finding eigenvalues involves solving:

• Ax = b for random b
• det(A – λI) = 0
• Trace(A) = 0
• Sum of rows equals zero

Correct Answer: det(A – λI) = 0

Q49. Which of the following is true about adding matrices?

• Matrix addition is defined only for square matrices
• Matrix addition is commutative and associative when dimensions match
• A + B = AB
• Addition changes matrix dimensions

Correct Answer: Matrix addition is commutative and associative when dimensions match

Q50. When using matrices to solve multiple linear reactions in pharmaceutics, why is checking rank important?

• Rank tells us nothing about solutions
• Rank indicates whether independent equations determine a unique solution
• Higher rank always means more errors
• Rank equals number of chemicals always

Correct Answer: Rank indicates whether independent equations determine a unique solution

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