Matrix multiplication MCQs With Answer

Matrix multiplication MCQs With Answer is a focused study resource designed for B. Pharm students to master matrix operations used in pharmacokinetics, formulation modeling, and lab data analysis. This practice set covers core concepts like dimension compatibility, associative and distributive laws, non-commutativity, identity and zero matrices, inverse criteria, and applications in compartmental models and linear systems relevant to pharmacy. Clear explanations and targeted questions help improve calculation speed and conceptual clarity for dosage optimization, reaction networks, and experimental design. Ideal for exam preparation, assignments, and practical problem solving, these MCQs reinforce both theory and application. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the necessary condition for multiplying two matrices A (m×n) and B (p×q)?

• m = p
• n = p
• n = q
• m = q

Correct Answer: n = p

Q2. If A is a 2×3 matrix and B is a 3×4 matrix, what is the dimension of AB?

• 2×3
• 3×4
• 2×4
• 3×2

Correct Answer: 2×4

Q3. Which property does matrix multiplication always satisfy for compatible matrices?

• Commutativity (AB = BA)
• Associativity ((AB)C = A(BC))
• Elementwise multiplication
• Symmetry (A = Aᵀ)

Correct Answer: Associativity ((AB)C = A(BC))

Q4. For square matrices, which matrix acts as the multiplicative identity?

• Zero matrix
• Diagonal matrix with zeros
• Identity matrix I
• Permutation matrix

Correct Answer: Identity matrix I

Q5. Is matrix multiplication commutative in general?

• Yes, always
• No, never
• Only for square matrices
• Only for certain matrices that satisfy AB = BA

Correct Answer: Only for certain matrices that satisfy AB = BA

Q6. The (i,j) entry of the product AB is computed by:

• Multiplying A(i,j) by B(i,j)
• Dot product of the i-th row of A with the j-th column of B
• Sum of i-th column of A and j-th row of B
• Cross product of row and column vectors

Correct Answer: Dot product of the i-th row of A with the j-th column of B

Q7. If A is 3×3 and invertible, which statement is true about AB = AC?

• It implies B = C only if A is invertible
• It always implies B = C
• It never implies B = C
• It implies B = C only if B and C are diagonal

Correct Answer: It implies B = C only if A is invertible

Q8. Which of the following is a reason matrix multiplication is used in pharmacokinetics?

• To perform elementwise drug concentration subtraction
• To combine compartment transfer rates with state vectors
• To compute pH levels directly
• To replace all differential equations

Correct Answer: To combine compartment transfer rates with state vectors

Q9. If A = [[1,2],[3,4]] and B = [[0,1],[1,0]], what is AB? (Write rows as comma-separated)

• [[2,1],[4,3]]
• [[2,1],[4,3]]
• [[2,1],[4,3]]
• [[1,0],[0,1]]

Correct Answer: [[2,1],[4,3]]

Q10. Which statement about the zero matrix 0 satisfies A0 = 0 for any conformable A?

• Only true for square A
• True for any conformable A
• Only true for invertible A
• False for nonzero A

Correct Answer: True for any conformable A

Q11. If A is m×n and B is n×m, what can be said about AB and BA?

• AB and BA are always equal
• AB is m×m and BA is n×n
• AB and BA are both m×n
• AB is n×n and BA is m×m

Correct Answer: AB is m×m and BA is n×n

Q12. Which property holds: A(B + C) = ?

• AB + AC (left distributive)
• BA + CA (wrong order)
• (B + C)A
• Only AB if C = 0

Correct Answer: AB + AC (left distributive)

Q13. For a scalar k and matrices A and B (compatible), which is true?

• k(AB) = (kA)B = A(kB)
• k(AB) = (kA)(kB)
• k(AB) = (A)(B)/k
• k(AB) = AB + k

Correct Answer: k(AB) = (kA)B = A(kB)

Q14. If A is 2×2 with rows r1,r2 and B is 2×2 with columns c1,c2, which entry is r2·c1?

• Entry (1,1)
• Entry (2,1)
• Entry (1,2)
• Entry (2,2)

Correct Answer: Entry (2,1)

Q15. A times identity I equals A. This demonstrates which property?

• Distributivity
• Associativity
• Existence of multiplicative identity
• Commutativity

Correct Answer: Existence of multiplicative identity

Q16. If det(A) ≠ 0 for square A, what does this imply about A?

• A is singular
• A is invertible
• A has dependent rows
• A has zero eigenvalues only

Correct Answer: A is invertible

Q17. Which of the following operations is used to compute product AB efficiently in small matrices?

• Elementwise multiplication
• Row-by-column dot products
• Column concatenation
• Inversion then multiplication

Correct Answer: Row-by-column dot products

Q18. In systems of linear equations Ax = b, why is matrix multiplication important?

• It represents combining coefficients with variables to produce constants
• It eliminates the need for any computations
• It changes the solution space dimension arbitrarily
• It always yields a scalar

Correct Answer: It represents combining coefficients with variables to produce constants

Q19. For diagonal matrix D and any matrix A (compatible), which is generally true?

• DA = AD always
• DA = A if D is zero
• DA scales rows (if D on left) or columns (if on right)
• DA equals the inverse of A

Correct Answer: DA scales rows (if D on left) or columns (if on right)

Q20. If A is 2×2 and B is 2×2, which equality is always false unless special conditions hold?

• det(AB) = det(A)det(B)
• (AB)ᵀ = BᵀAᵀ
• AB = BA
• Trace(AB) = Trace(BA)

Correct Answer: AB = BA

Q21. In pharmacology modeling, multiplying a transition matrix by a state vector yields:

• A scalar concentration
• A new state vector of compartment amounts
• The determinant of the system
• The eigenvalues directly

Correct Answer: A new state vector of compartment amounts

Q22. If A is m×n and rank(A) = r, what is the maximum rank of AB (B conformable)?

• min(r, rank(B))
• max(r, rank(B))
• r + rank(B)
• m + n

Correct Answer: min(r, rank(B))

Q23. For two matrices A (2×3) and B (3×2), what is true about trace(AB) compared to trace(BA)?

• trace(AB) is undefined
• trace(AB) = trace(BA)
• trace(AB) = -trace(BA)
• trace(AB) equals determinant of A

Correct Answer: trace(AB) = trace(BA)

Q24. Multiplying a matrix by its inverse yields:

• Zero matrix
• Identity matrix
• Transpose of the matrix
• Negative identity

Correct Answer: Identity matrix

Q25. Which multiplication technique helps when matrices have many zeros (sparse)?

• Standard dense multiplication
• Exploiting sparsity to skip zero products
• Converting to full matrices first
• Using determinant formulas

Correct Answer: Exploiting sparsity to skip zero products

Q26. If A is 3×3 and B is 3×3, and AB = I, what can we conclude?

• B is left-inverse of A and A is invertible with B = A⁻¹
• Only that A is singular
• Only that B is singular
• No conclusion about invertibility

Correct Answer: B is left-inverse of A and A is invertible with B = A⁻¹

Q27. The product of two upper triangular matrices is:

• Lower triangular
• Upper triangular
• Diagonal always
• Full matrix with no structure

Correct Answer: Upper triangular

Q28. For block matrices, which rule can simplify multiplication?

• Treat as scalar multiplication only
• Use partitioned multiplication following sub-block compatibility
• Blocks must be diagonal to multiply
• Block multiplication is undefined

Correct Answer: Use partitioned multiplication following sub-block compatibility

Q29. Which of the following is true about (A + B)² for matrices?

• (A + B)² = A² + 2AB + B² always
• (A + B)² = A² + AB + BA + B²
• (A + B)² = A² + B² only
• (A + B)² = A² – AB + B²

Correct Answer: (A + B)² = A² + AB + BA + B²

Q30. If Aᵀ is the transpose of A, what is (AB)ᵀ equal to?

• AᵀBᵀ
• BᵀAᵀ
• Aᵀ + Bᵀ
• ABᵀ

Correct Answer: BᵀAᵀ

Q31. Which calculation is most relevant when using matrices to fit linear regression for assay calibration?

• Solving XᵀXβ = Xᵀy using matrix multiplication
• Computing elementwise division only
• Finding the inverse of y directly
• Multiplying residuals by zeros

Correct Answer: Solving XᵀXβ = Xᵀy using matrix multiplication

Q32. In practice, why is order important when multiplying matrices representing sequential processes?

• Order does not matter for sequential processes
• Different orders change meaning and results of transformations
• Changing order always yields the same determinant
• Order only matters for scalars

Correct Answer: Different orders change meaning and results of transformations

Q33. If A is 2×3 and B is 3×1, what shape is AB and what does it represent in linear systems?

• 2×1, a vector of linear combinations of columns of A
• 3×3, a square matrix
• 2×3, same as A
• Unknown shape

Correct Answer: 2×1, a vector of linear combinations of columns of A

Q34. In numerical computations, which factor can lead to inaccuracy in matrix multiplication results?

• Floating-point round-off errors
• Exact arithmetic always
• Using integer-only operations
• Multiplying with identity matrices only

Correct Answer: Floating-point round-off errors

Q35. The Frobenius norm of product AB is bounded by which inequality?

• ||AB||_F ≤ ||A||_F ||B||_F
• ||AB||_F = ||A||_F + ||B||_F
• ||AB||_F ≥ ||A||_F ||B||_F
• No relation exists

Correct Answer: ||AB||_F ≤ ||A||_F ||B||_F

Q36. When modeling drug interactions in two compartments, a 2×2 transfer matrix multiplied by a 2×1 state vector yields:

• New compartment amounts after transfer
• Only the total mass, not distribution
• The inverse transfer rates
• A scalar elimination rate

Correct Answer: New compartment amounts after transfer

Q37. Which of the following can prevent AB from being defined even if A and B are square?

• Different element types
• They are square so AB is always defined
• If their sizes differ (e.g., 2×2 and 3×3)
• If determinants are zero

Correct Answer: If their sizes differ (e.g., 2×2 and 3×3)

Q38. If A is orthogonal (AᵀA = I), what is A⁻¹ equal to?

• Aᵀ
• -A
• Determinant of A
• Zero matrix

Correct Answer: Aᵀ

Q39. Which scenario uses matrix multiplication to combine experimental replicates into predicted responses?

• Multiplying design matrix by parameter vector
• Adding matrices elementwise only
• Computing determinants of each replicate
• Using scalar division per replicate

Correct Answer: Multiplying design matrix by parameter vector

Q40. What is the result of multiplying a 1×n row vector by an n×1 column vector?

• A 1×1 scalar (dot product)
• An n×n matrix
• A 1×n vector
• An n×1 vector

Correct Answer: A 1×1 scalar (dot product)

Q41. Which best describes computational complexity of naive matrix multiplication for n×n matrices?

• O(n)
• O(n^2)
• O(n^3)
• O(2^n)

Correct Answer: O(n^3)

Q42. If two matrices A and B are diagonalizable and share the same eigenvectors, what can be said about AB?

• AB is not diagonalizable
• AB is diagonalizable and eigenvectors are the same
• AB must be zero
• AB is always symmetric

Correct Answer: AB is diagonalizable and eigenvectors are the same

Q43. When performing least squares, which multiplication is central to normal equations?

• X Xᵀ y = β
• Xᵀ X β = Xᵀ y
• yᵀ X X = β
• X β = 0

Correct Answer: Xᵀ X β = Xᵀ y

Q44. If A is singular, what is true about multiplication by A when solving Ax = b?

• Unique solution always exists
• May have no solution or infinitely many solutions
• Always has exactly two solutions
• Determinant of A will be nonzero

Correct Answer: May have no solution or infinitely many solutions

Q45. A 3×3 rotation matrix R multiplied by vector v rotates v. Which property of R preserves vector length?

• R is orthogonal with determinant ±1
• R is singular
• R is diagonal with different scales
• R has zero trace

Correct Answer: R is orthogonal with determinant ±1

Q46. In experimental design, a contrast matrix C multiplied by treatment means μ gives:

• Predicted contrasts (linear combinations) of means
• Determinants of treatments
• Inverse of treatment effects
• Only error terms

Correct Answer: Predicted contrasts (linear combinations) of means

Q47. Which is a valid approach to check correctness of a computed matrix product AB?

• Compare AB to BA always
• Verify a few entries using row-by-column dot products
• Compute determinants and expect them equal
• Check that AB equals A + B

Correct Answer: Verify a few entries using row-by-column dot products

Q48. If A has dimensions 4×2 and B has dimensions 2×5, how many scalar multiplications to compute AB naively?

• 40
• 4×2×5 = 40 scalar multiplications
• 8
• 10

Correct Answer: 4×2×5 = 40 scalar multiplications

Q49. Which statement is true about singular value decomposition (SVD) and multiplication?

• SVD cannot be used with multiplication
• SVD represents A as UΣVᵀ and helps analyze products
• SVD always gives integer matrices
• SVD equals the determinant of A

Correct Answer: SVD represents A as UΣVᵀ and helps analyze products

Q50. In practice, why combine matrices when modeling multi-step drug processing pipelines?

• To obscure individual step effects
• To produce a single transformation representing sequential steps
• To reduce all data to scalars only
• Because multiplication always commutes

Correct Answer: To produce a single transformation representing sequential steps

Download