Matrices – Introduction MCQs With Answer

Matrices – Introduction MCQs With Answer provides B. Pharm students a focused primer on matrix theory and its pharmaceutical applications. This concise guide covers matrix types, operations, determinants, inverses, rank, eigenvalues, and solving linear systems—key tools for pharmacokinetics, compartmental models, dosage optimization, and calibration problems. Emphasizing practical examples and problem-solving strategies, these MCQs reinforce conceptual understanding and computational skills essential for quantitative pharmacy courses. SEO keywords included: matrices, matrix operations, inverse matrix, determinants, linear systems, pharmacokinetics, B. Pharm, MCQs with answers. Clear explanations help bridge math fundamentals to real-world pharmaceutical calculations. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is a matrix?

• A rectangular array of numbers arranged in rows and columns
• A single algebraic equation
• A scalar quantity representing concentration
• A function mapping compounds to reactions

Correct Answer: A rectangular array of numbers arranged in rows and columns

Q2. Which of the following is a square matrix?

• 2×3 matrix
• 3×3 matrix
• 1×4 matrix
• 3×2 matrix

Correct Answer: 3×3 matrix

Q3. What is the identity matrix I for 3×3 matrices?

• A matrix with all entries one
• A diagonal matrix with zeros on diagonal
• A diagonal matrix with ones on diagonal and zeros elsewhere
• A matrix with ones above the diagonal only

Correct Answer: A diagonal matrix with ones on diagonal and zeros elsewhere

Q4. How is matrix addition defined?

• Element-wise addition for matrices of same order
• Multiplying corresponding entries
• Appending rows of one to another
• Only for square matrices by determinant sum

Correct Answer: Element-wise addition for matrices of same order

Q5. When is matrix multiplication AB defined?

• Number of rows of A equals number of rows of B
• Number of columns of A equals number of rows of B
• Both A and B are square
• When determinants of A and B are nonzero

Correct Answer: Number of columns of A equals number of rows of B

Q6. Which property does matrix multiplication generally NOT satisfy?

• Associativity
• Distributivity over addition
• Commutativity
• Compatibility with scalar multiplication

Correct Answer: Commutativity

Q7. What is the transpose of a matrix A?

• A matrix obtained by multiplying each entry by −1
• A matrix obtained by reflecting A across its main diagonal
• The inverse of A
• A matrix with rows and columns swapped and entries negated

Correct Answer: A matrix obtained by reflecting A across its main diagonal

Q8. Which matrix is symmetric?

• A^T = A
• A^T = −A
• A is diagonalizable only
• A has zero determinant

Correct Answer: A^T = A

Q9. What defines a skew-symmetric matrix?

• A^T = A
• A^T = −A
• All diagonal elements are one
• It is singular

Correct Answer: A^T = −A

Q10. How do you compute the determinant of a 2×2 matrix [[a,b],[c,d]]?

• ad − bc
• ab + cd
• ac − bd
• a + d

Correct Answer: ad − bc

Q11. A 2×2 matrix is invertible if:

• Its determinant is zero
• Its determinant is nonzero
• It is symmetric
• It has at least one zero entry

Correct Answer: Its determinant is nonzero

Q12. What is the inverse of a 2×2 matrix [[a,b],[c,d]] (when invertible)?

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

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

Q13. What is the rank of a matrix?

• The number of zero rows only
• The maximum number of linearly independent rows or columns
• The trace of the matrix
• The determinant of the matrix

Correct Answer: The maximum number of linearly independent rows or columns

Q14. Which statement about rank is true?

• Row rank always equals column rank
• Row rank is always greater than column rank
• Column rank is determinant-dependent
• Rank applies only to square matrices

Correct Answer: Row rank always equals column rank

Q15. What does an augmented matrix represent?

• A matrix of coefficients only
• A matrix including coefficients and constants for a linear system
• A diagonal representation of variables
• An inverse matrix appended to the original

Correct Answer: A matrix including coefficients and constants for a linear system

Q16. Which method uses elementary row operations to solve linear systems?

• Cramer’s rule
• Gaussian elimination
• Eigen decomposition
• Matrix inversion only

Correct Answer: Gaussian elimination

Q17. Cramer’s rule requires what condition on the coefficient matrix?

• It must be symmetric
• Determinant must be nonzero (matrix invertible)
• All entries positive
• Matrix must be diagonal

Correct Answer: Determinant must be nonzero (matrix invertible)

Q18. In pharmacokinetics, compartment models sometimes use matrices to:

• Simplify dispensing labels
• Model transfer rates between compartments
• Replace chemical assays
• Visualize molecular structures

Correct Answer: Model transfer rates between compartments

Q19. Which matrix property ensures A A^−1 = I?

• A is singular
• A is invertible
• A is skew-symmetric
• A has zero trace

Correct Answer: A is invertible

Q20. The trace of a square matrix is defined as:

• Sum of all entries
• Sum of diagonal entries
• Product of diagonal entries
• Number of nonzero rows

Correct Answer: Sum of diagonal entries

Q21. What are eigenvalues of a matrix A?

• Scalars λ satisfying A v = λ v for some nonzero vector v
• Determinants of submatrices
• Only diagonal entries of A
• Inverse elements of A

Correct Answer: Scalars λ satisfying A v = λ v for some nonzero vector v

Q22. In linear algebra, eigenvectors correspond to:

• Vectors that change direction under A
• Vectors whose direction is unchanged by A, only scaled
• Only zero vectors
• Vectors orthogonal to all columns

Correct Answer: Vectors whose direction is unchanged by A, only scaled

Q23. Which decomposition factors a matrix into lower and upper triangular matrices?

• SVD (Singular Value Decomposition)
• LU decomposition
• QR decomposition
• Eigen decomposition

Correct Answer: LU decomposition

Q24. How is the determinant related to volume in linear transformations?

• Determinant scales volume by factor equal to its absolute value
• Determinant equals the new volume directly
• Determinant has no geometric meaning
• Determinant inverts volume change

Correct Answer: Determinant scales volume by factor equal to its absolute value

Q25. What is a null space (kernel) of a matrix A?

• Set of vectors x such that A x = 0
• Set of solutions to A x = b for nonzero b
• Space spanned by columns of A
• Inverse of column space

Correct Answer: Set of vectors x such that A x = 0

Q26. If a 3×3 matrix has determinant zero, what can you conclude?

• It is invertible
• It is singular and not invertible
• It has three independent columns
• Its trace must be zero

Correct Answer: It is singular and not invertible

Q27. What does orthogonal matrix Q satisfy?

• Q^T Q = I
• Q^2 = I always
• Q has zero determinant
• Q is diagonal only

Correct Answer: Q^T Q = I

Q28. Which method is efficient for solving large sparse linear systems common in compartmental models?

• Direct inversion of full matrix
• Iterative methods such as conjugate gradient
• Manual Gaussian elimination only
• Cramer’s rule

Correct Answer: Iterative methods such as conjugate gradient

Q29. What is the effect of multiplying a matrix by a scalar k?

• Only the determinant changes
• Each entry of the matrix is multiplied by k
• Rows and columns are swapped
• Matrix becomes singular

Correct Answer: Each entry of the matrix is multiplied by k

Q30. In regression and calibration, which matrix expression is used for least squares parameter estimate?

• (X^T X)^−1 X^T y
• X X^T y
• X^T y only
• Determinant of X times y

Correct Answer: (X^T X)^−1 X^T y

Q31. What defines a diagonal matrix?

• All non-diagonal entries are zero
• All entries are equal
• It has nonzero entries only above diagonal
• Columns sum to zero

Correct Answer: All non-diagonal entries are zero

Q32. Which matrix has nonzero elements only on the main diagonal and possibly one side diagonal?

• Full matrix
• Banded or tridiagonal matrix
• Permutation matrix
• Zero matrix

Correct Answer: Banded or tridiagonal matrix

Q33. What is a permutation matrix used for in Gaussian elimination?

• To scale rows by constants
• To interchange rows (pivoting)
• To add multiple of one row to another
• To compute inverse directly

Correct Answer: To interchange rows (pivoting)

Q34. If columns of matrix A are linearly dependent, then:

• A has full column rank
• Determinant of A (if square) is zero
• Null space contains only trivial vector
• A is orthogonal

Correct Answer: Determinant of A (if square) is zero

Q35. What is a block matrix?

• A matrix partitioned into submatrices
• A matrix with only zeros and ones
• A diagonal matrix with blocks of zeros
• A symmetric matrix only

Correct Answer: A matrix partitioned into submatrices

Q36. Inverse of a product of invertible matrices AB is:

• A^−1 B^−1
• B^−1 A^−1
• (AB)^−1 = AB
• Undefined for square matrices

Correct Answer: B^−1 A^−1

Q37. Which operation can be used to find whether columns are independent?

• Compute matrix transpose only
• Perform row reduction to echelon form and check pivot columns
• Compute product with identity
• Calculate trace

Correct Answer: Perform row reduction to echelon form and check pivot columns

Q38. What is the determinant of the identity matrix I_n?

• 0
• n
• 1
• Depends on n! (factorial)

Correct Answer: 1

Q39. Which of the following is true about singular matrices?

• They always have an inverse
• Their determinant equals zero
• They are always diagonalizable
• They must be symmetric

Correct Answer: Their determinant equals zero

Q40. How can matrices help in multi-compartment pharmacokinetic models?

• They store only labels of compartments
• They represent transfer rate constants and compute concentration vectors over time
• Matrices are not applicable to PK models
• They replace chemical equilibrium constants

Correct Answer: They represent transfer rate constants and compute concentration vectors over time

Q41. What does the spectral radius of a matrix refer to?

• Maximum absolute value of its eigenvalues
• Number of nonzero rows
• Sum of eigenvectors
• Determinant magnitude always

Correct Answer: Maximum absolute value of its eigenvalues

Q42. Which decomposition is best for computing principal components or reducing dimensionality?

• LU decomposition
• SVD (Singular Value Decomposition)
• Cramer’s rule
• Determinant factoring

Correct Answer: SVD (Singular Value Decomposition)

Q43. If A is m×n and rank(A) = r, what is dimension of column space?

• m + n − r
• r
• n − r
• m × n

Correct Answer: r

Q44. How is an augmented matrix reduced to find system consistency?

• By determinant expansion only
• By row-reduction to reduced row echelon form (RREF)
• By computing trace and comparing
• By transposing and inverting only

Correct Answer: By row-reduction to reduced row echelon form (RREF)

Q45. Which of the following indicates a unique solution to a linear system Ax = b?

• Coefficient matrix A singular
• Coefficient matrix A invertible (full rank)
• Augmented matrix has inconsistent row
• Null space dimension > 0

Correct Answer: Coefficient matrix A invertible (full rank)

Q46. In context of drug formulation, matrices can be applied to:

• Only to label bottles
• Optimize formulations via mixture design and multivariate calibration
• Replace wet-lab stability studies
• Calculate molecular weights only

Correct Answer: Optimize formulations via mixture design and multivariate calibration

Q47. Which statement about eigenvectors corresponding to distinct eigenvalues is true?

• They are linearly dependent
• They are linearly independent
• They must be orthogonal always
• They are all zero vectors

Correct Answer: They are linearly independent

Q48. What is the effect of pre-multiplying a vector by a matrix A?

• It scales the vector element-wise
• It applies a linear transformation mapping the vector to a new vector in row-space
• It transposes the vector only
• It computes the determinant with the vector

Correct Answer: It applies a linear transformation mapping the vector to a new vector in row-space

Q49. Which matrix norm is equal to the maximum absolute column sum?

• Frobenius norm
• Infinity norm (maximum absolute row sum)
• One norm (maximum absolute column sum)
• Spectral norm

Correct Answer: One norm (maximum absolute column sum)

Q50. For solving Ax = b numerically in pharmaceutical modeling, best practices include:

• Ignoring conditioning and using naive inversion
• Checking matrix conditioning, using stable algorithms (LU/QR/SVD) and validating results
• Always using Cramer’s rule regardless of size
• Only using hand calculations

Correct Answer: Checking matrix conditioning, using stable algorithms (LU/QR/SVD) and validating results

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