Types of matrices MCQs With Answer

Types of matrices MCQs With Answer is an essential resource for B.Pharm students preparing for exams in pharmaceutical mathematics and applied quantitative methods. This guide covers matrix classification—such as square, diagonal, identity, triangular, symmetric, skew-symmetric, singular, orthogonal and sparse matrices—along with properties like determinant, rank, inverse and transpose, and their relevance in formulation design, compartmental pharmacokinetics and stability analysis. Each MCQ is crafted to reinforce concepts, improve problem-solving speed and build exam confidence. The questions emphasize definitions, computations and practical applications relevant to pharmacy practice. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What defines a square matrix?

• A matrix with same number of rows and columns
• A matrix with more rows than columns
• A matrix with all zero entries
• A matrix with only one non-zero entry

Correct Answer: A matrix with same number of rows and columns

Q2. Which matrix has nonzero entries only on its main diagonal?

• Triangular matrix
• Diagonal matrix
• Identity matrix
• Zero matrix

Correct Answer: Diagonal matrix

Q3. Which property does an identity matrix I have?

• I + I = 2I only for 2×2 matrices
• Multiplying any compatible matrix by I leaves it unchanged
• I has nonzero off-diagonal entries
• I is always singular

Correct Answer: Multiplying any compatible matrix by I leaves it unchanged

Q4. A matrix whose transpose equals itself is called:

• Skew-symmetric
• Orthogonal
• Symmetric
• Singular

Correct Answer: Symmetric

Q5. A matrix A is skew-symmetric if:

• A^T = A
• A^T = -A
• AA^T = I
• det(A) = 0

Correct Answer: A^T = -A

Q6. Which matrix has all elements equal to zero?

• Identity matrix
• Zero (null) matrix
• Diagonal matrix
• Permutation matrix

Correct Answer: Zero (null) matrix

Q7. A triangular matrix is one that has:

• Non-zero entries only above or below the main diagonal
• Non-zero entries only on the main diagonal
• All rows identical
• Only one non-zero column

Correct Answer: Non-zero entries only above or below the main diagonal

Q8. Which matrix type is useful for representing discrete compartmental transitions in pharmacokinetics?

• Sparse matrix
• Stochastic (transition) matrix
• Diagonal matrix
• Hankel matrix

Correct Answer: Stochastic (transition) matrix

Q9. A matrix is singular when:

• Its determinant is zero
• Its determinant is one
• It has full rank
• It is orthogonal

Correct Answer: Its determinant is zero

Q10. The inverse of a nonsingular matrix A satisfies:

• A * A^T = I
• A * A^{-1} = I
• A^{-1} = det(A)
• A^{-1} = A^2

Correct Answer: A * A^{-1} = I

Q11. Rank of a matrix corresponds to:

• The number of nonzero rows after row reduction
• Number of columns only
• Determinant value
• Trace of the matrix

Correct Answer: The number of nonzero rows after row reduction

Q12. Trace of a square matrix is defined as:

• Sum of all elements
• Sum of diagonal elements
• Product of diagonal elements
• Determinant of the matrix

Correct Answer: Sum of diagonal elements

Q13. An orthogonal matrix Q satisfies which relationship?

• Q^T Q = I
• Q^T = -Q
• det(Q) = 0 only
• Q is diagonal

Correct Answer: Q^T Q = I

Q14. A matrix with at least one zero row or column always has:

• Full rank
• Rank less than maximum
• Nonzero determinant
• Orthogonal columns

Correct Answer: Rank less than maximum

Q15. Idempotent matrices satisfy which condition?

• A^2 = 0
• A^2 = A
• A^T = A^{-1}
• A is singular only if det(A)=1

Correct Answer: A^2 = A

Q16. Which matrix type often appears in finite difference discretizations and has most entries zero?

• Dense matrix
• Sparse matrix
• Full rank matrix
• Hankel matrix

Correct Answer: Sparse matrix

Q17. A block matrix is:

• A matrix divided into submatrices or blocks
• A matrix with negative determinant
• A matrix with equal rows
• A diagonal matrix with scalar entries

Correct Answer: A matrix divided into submatrices or blocks

Q18. A band matrix is characterized by:

• Nonzero entries concentrated near the main diagonal
• Nonzero entries only on the corners
• All entries equal
• Only one row nonzero

Correct Answer: Nonzero entries concentrated near the main diagonal

Q19. Which matrix type has constant values along each descending diagonal from left to right?

• Toeplitz matrix
• Hankel matrix
• Permutation matrix
• Identity matrix

Correct Answer: Toeplitz matrix

Q20. A Hankel matrix has constant values along:

• Rows only
• Columns only
• Anti-diagonals (ascending diagonals)
• Main diagonal

Correct Answer: Anti-diagonals (ascending diagonals)

Q21. Which matrix property is essential to determine if linear equations have a unique solution?

• Trace equals zero
• Matrix is singular
• Matrix is nonsingular (invertible)
• Matrix is triangular

Correct Answer: Matrix is nonsingular (invertible)

Q22. The determinant of a triangular matrix equals:

• Sum of diagonal entries
• Product of diagonal entries
• Zero always
• Product of off-diagonal entries

Correct Answer: Product of diagonal entries

Q23. Which operation changes rows into columns?

• Inverse
• Transpose
• Determinant
• Trace

Correct Answer: Transpose

Q24. For which matrix does A A^T = A^T A hold generally?

• Only diagonal matrices
• Only symmetric matrices
• For any matrix (commutative property)
• For matrices that commute with their transpose, such as symmetric or orthogonal

Correct Answer: For matrices that commute with their transpose, such as symmetric or orthogonal

Q25. In pharmaceutical modeling, a compartmental model represented by a system of linear ODEs uses what matrix form?

• Stochastic matrix for drug absorption only
• Coefficient matrix governing transfer rates between compartments
• Diagonal matrix with zeros off-diagonal only
• Hankel matrix for concentration-time curves

Correct Answer: Coefficient matrix governing transfer rates between compartments

Q26. A permutation matrix is obtained by:

• Permuting rows or columns of an identity matrix
• Rotating a diagonal matrix
• Multiplying two zero matrices
• Taking inverse of a triangular matrix

Correct Answer: Permuting rows or columns of an identity matrix

Q27. A matrix with orthonormal columns implies:

• Columns are linearly dependent
• Q^T Q = I for the matrix Q of columns
• Determinant must be zero
• All diagonal entries are equal

Correct Answer: Q^T Q = I for the matrix Q of columns

Q28. The null space (kernel) of a matrix consists of:

• Vectors mapped to zero by the matrix
• Vectors mapped to identity
• Only eigenvectors with positive eigenvalues
• All columns of the matrix

Correct Answer: Vectors mapped to zero by the matrix

Q29. A matrix A is positive definite if:

• x^T A x > 0 for all nonzero vectors x
• det(A) < 0 always
• A has only integer entries
• A is singular

Correct Answer: x^T A x > 0 for all nonzero vectors x

Q30. Which matrix is commonly used to normalize vectors and preserve length?

• Singular matrix
• Orthogonal matrix
• Nilpotent matrix
• Idempotent matrix

Correct Answer: Orthogonal matrix

Q31. Nilpotent matrices satisfy which condition?

• A^k = 0 for some positive integer k
• A^2 = A always
• det(A) = 1
• A is orthogonal

Correct Answer: A^k = 0 for some positive integer k

Q32. Which of the following is a scalar matrix?

• A diagonal matrix with all diagonal entries equal to the same scalar
• A triangular matrix with different diagonal entries
• A matrix with only zeros and ones randomly placed
• A Toeplitz matrix

Correct Answer: A diagonal matrix with all diagonal entries equal to the same scalar

Q33. The adjoint (classical adjugate) of a matrix is used to compute:

• Trace
• Inverse for nonsingular matrices
• Rank only
• Transpose

Correct Answer: Inverse for nonsingular matrices

Q34. Which statement about row echelon form (REF) is true?

• All pivot entries are to the right of the pivot in the previous row
• REF guarantees orthogonality of rows
• REF equals reduced row echelon form always
• REF increases the determinant

Correct Answer: All pivot entries are to the right of the pivot in the previous row

Q35. For a 3×3 matrix, if two rows are identical, then:

• Matrix is invertible
• Rank is 3
• Determinant is zero
• Matrix is orthogonal

Correct Answer: Determinant is zero

Q36. Which matrix is characterized by having only one non-zero element in each row and column and determinant ±1?

• Permutation matrix
• Zero matrix
• Diagonal matrix with zeros
• Hankel matrix

Correct Answer: Permutation matrix

Q37. What is a stochastic matrix used to model?

• Conservation of mass only
• Probability transitions where each column or row sums to one
• Matrices with negative eigenvalues only
• Symmetric interactions only

Correct Answer: Probability transitions where each column or row sums to one

Q38. If A is invertible and k is a scalar, (kA)^{-1} equals:

• k^{-1} A^{-1}
• k A^{-1}
• A^{-1} regardless of k
• k^n A for n×n matrix

Correct Answer: k^{-1} A^{-1}

Q39. Which statement about eigenvalues is correct?

• Eigenvalues are always positive for any matrix
• Eigenvalues λ satisfy det(A – λI) = 0
• Eigenvalues equal the diagonal entries for all matrices
• Eigenvalues are only defined for diagonal matrices

Correct Answer: Eigenvalues λ satisfy det(A – λI) = 0

Q40. A singular value decomposition (SVD) represents a matrix as product of:

• Three matrices including orthogonal and diagonal matrices
• Only triangular matrices
• Sum of two zero matrices
• A diagonal matrix only

Correct Answer: Three matrices including orthogonal and diagonal matrices

Q41. Which matrix operation is not generally commutative?

• Addition of matrices
• Scalar multiplication
• Matrix multiplication
• Transposition

Correct Answer: Matrix multiplication

Q42. The condition number of a matrix provides information about:

• How singular a matrix is and numerical sensitivity of solutions
• Only the determinant value
• Whether the matrix is diagonal
• Trace magnitude only

Correct Answer: How singular a matrix is and numerical sensitivity of solutions

Q43. In solving Ax = b, if A is lower triangular, x can be found by:

• Backward substitution
• Forward substitution
• Eigen decomposition
• Computing the trace

Correct Answer: Forward substitution

Q44. A Gram matrix is formed by taking inner products of vectors; it is always:

• Skew-symmetric
• Positive semidefinite
• Singular only
• Permutation matrix

Correct Answer: Positive semidefinite

Q45. The Moore-Penrose pseudoinverse is useful when:

• Matrix is square and invertible only
• Solving least-squares problems for non-square or singular matrices
• Computing the trace
• Finding eigenvalues only

Correct Answer: Solving least-squares problems for non-square or singular matrices

Q46. Which matrix type preserves the Euclidean norm of vectors when multiplied?

• Idempotent matrix
• Orthogonal matrix
• Nilpotent matrix
• Singular matrix

Correct Answer: Orthogonal matrix

Q47. If A is symmetric positive definite, which decomposition is commonly used for numerical stability?

• LU decomposition without pivoting
• Cholesky decomposition
• QR decomposition of A^T only
• Hankel decomposition

Correct Answer: Cholesky decomposition

Q48. Which matrix has determinant equal to the permutation sign for each permutation matrix?

• Identity matrix only
• Permutation matrix
• Zero matrix
• Diagonal matrix with unequal entries

Correct Answer: Permutation matrix

Q49. The Frobenius norm of a matrix is computed as:

• The maximum absolute column sum
• The square root of sum of squares of all entries
• The product of eigenvalues only
• The sum of singular values squared only

Correct Answer: The square root of sum of squares of all entries

Q50. In linear regression design, the matrix X’X must be invertible to:

• Guarantee unique least-squares estimates of coefficients
• Ensure residuals sum to one
• Produce orthogonal predictors only
• Make the model nonlinear

Correct Answer: Guarantee unique least-squares estimates of coefficients

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