Singular and Non-singular matrices MCQs With Answer

Singular and Non-singular matrices MCQs With Answer

Understanding singular and non-singular matrices is essential for B. Pharm students who apply linear algebra in pharmacokinetics, drug formulation models, and data analysis. This concise introduction covers determinants, invertibility, rank, null space, and practical criteria to identify singular (determinant zero, non-invertible) versus non-singular matrices (non-zero determinant, invertible). Emphasis is on computation methods, row operations, 2×2 inverse formula, and implications for solving linear systems used in compartmental modeling and dose optimization. Clear concepts help ensure unique solutions in linear systems arising in pharmacy problems. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the defining property of a non-singular matrix?

• Its determinant is zero
• It has no inverse
• Its determinant is non-zero
• It has at least one zero eigenvalue

Correct Answer: Its determinant is non-zero

Q2. Which condition indicates a matrix is singular?

• Full column rank
• Non-zero determinant
• Determinant equal to zero
• Invertible linear transformation

Correct Answer: Determinant equal to zero

Q3. For a 2×2 matrix [[a,b],[c,d]], which formula gives the inverse when it is non-singular?

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

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

Q4. If a matrix has a zero eigenvalue, what can be concluded?

• Matrix is non-singular
• Matrix is singular
• Matrix determinant is non-zero
• Matrix is orthogonal

Correct Answer: Matrix is singular

Q5. How does a single row of zeros in a matrix affect singularity?

• Matrix must be non-singular
• Matrix is singular because rank is reduced
• Matrix determinant becomes one
• Matrix remains invertible if columns are independent

Correct Answer: Matrix is singular because rank is reduced

Q6. What is the relationship between matrix rank and non-singularity for an n×n matrix?

• Non-singular if rank < n
• Non-singular if rank = n
• Non-singular if rank = 0
• Rank does not determine singularity

Correct Answer: Non-singular if rank = n

Q7. Which row operation can change the determinant by a sign?

• Adding a multiple of one row to another
• Swapping two rows
• Multiplying a row by 1
• Replacing a row by itself

Correct Answer: Swapping two rows

Q8. If two rows of a square matrix are identical, what is the determinant?

• Non-zero
• Equal to one
• Zero
• Undefined

Correct Answer: Zero

Q9. For square matrices A and B of same size, det(AB) equals:

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

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

Q10. If det(A) = 0, then A is:

• Invertible
• Singular
• Orthogonal
• Diagonalizable only

Correct Answer: Singular

Q11. Which property ensures a unique solution to Ax = b for square A?

• A is singular
• A has zero determinant
• A is non-singular (invertible)
• A has dependent columns

Correct Answer: A is non-singular (invertible)

Q12. What is the null space of a non-singular n×n matrix?

• All of R^n
• A one-dimensional subspace
• {0} only
• Infinite-dimensional

Correct Answer: {0} only

Q13. Cramer’s rule requires which condition on the coefficient matrix?

• Matrix must be singular
• Matrix must be non-singular
• Matrix must be symmetric
• Matrix must be diagonal

Correct Answer: Matrix must be non-singular

Q14. Which of the following guarantees det(A^T) = det(A)?

• A is singular
• Transpose property holds for all square matrices
• Only for diagonal matrices
• Only for orthogonal matrices

Correct Answer: Transpose property holds for all square matrices

Q15. For an upper triangular square matrix, the determinant equals:

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

Correct Answer: Product of diagonal elements

Q16. If A is invertible, what is det(A^{-1}) in terms of det(A)?

• det(A) *
• 1/det(A)
• det(A)^2
• −det(A)

Correct Answer: 1/det(A)

Q17. Multiplying a single row of a matrix by scalar k changes the determinant by:

• Adding k to determinant
• Multiplying determinant by k
• Multiplying determinant by k^2
• No change

Correct Answer: Multiplying determinant by k

Q18. The adjugate (classical adjoint) of A helps compute:

• Eigenvalues
• Inverse of A when det(A) ≠ 0
• Rank directly
• Nullspace dimension

Correct Answer: Inverse of A when det(A) ≠ 0

Q19. If A is singular, which of the following is true about the homogeneous system Ax = 0?

• Only trivial solution exists
• No solutions exist
• Infinitely many solutions exist
• Exactly two solutions exist

Correct Answer: Infinitely many solutions exist

Q20. Which test can quickly show singularity for a 3×3 matrix?

• Compute trace
• Compute determinant
• Compute sum of rows
• Check if matrix is symmetric

Correct Answer: Compute determinant

Q21. If det(A) = 5 and det(B) = 2, det(2A B) for square A,B equals:

• det(2A) + det(B)
• 2^n * det(A) * det(B) where n is size
• det(A) * det(B) /2
• det(A) + det(B)

Correct Answer: 2^n * det(A) * det(B) where n is size

Q22. Which statement is true about eigenvalues of a non-singular matrix?

• At least one eigenvalue is zero
• All eigenvalues are non-zero
• All eigenvalues must be positive
• Eigenvalues equal diagonal entries only

Correct Answer: All eigenvalues are non-zero

Q23. Two matrices are inverses if their product equals:

• Zero matrix
• Identity matrix
• Diagonal matrix
• Transpose of one

Correct Answer: Identity matrix

Q24. If A has linearly dependent columns, then A is:

• Non-singular
• Singular
• Orthogonal
• Diagonalizable only

Correct Answer: Singular

Q25. The determinant of an orthogonal matrix is:

• Always zero
• Either +1 or −1
• Always positive and >1
• Equal to the trace

Correct Answer: Either +1 or −1

Q26. Which operation does NOT change linear dependence relations among columns?

• Swapping rows
• Adding a multiple of one row to another row
• Multiplying a row by zero
• Multiplying a column by a non-zero scalar

Correct Answer: Multiplying a column by a non-zero scalar

Q27. For matrices, if A is invertible and AB = AC, what follows?

• B = C
• B and C are singular
• det(B) = det(C) only
• No conclusion

Correct Answer: B = C

Q28. In pharmacokinetic compartment models represented by matrices, non-singularity ensures:

• No steady state exists
• Unique parameter estimates from linear equations
• Infinite parameter solutions always
• Model is non-linear

Correct Answer: Unique parameter estimates from linear equations

Q29. The determinant of a matrix equals zero when columns are:

• Linearly independent
• Linearly dependent
• Orthogonal
• Normalized

Correct Answer: Linearly dependent

Q30. Which of these implies A is invertible?

• Nullity(A) > 0
• Rank(A) = n for n×n matrix
• There exists non-zero x s.t. Ax = 0
• det(A) = 0

Correct Answer: Rank(A) = n for n×n matrix

Q31. If A and B are invertible, (A+B)^{-1} equals:

• A^{-1} + B^{-1}
• A^{-1}B^{-1}
• Not determined simply from A^{-1} and B^{-1}
• (A^{-1} + B^{-1})^{-1}

Correct Answer: Not determined simply from A^{-1} and B^{-1}

Q32. The determinant of the transpose of A is:

• Negative of det(A)
• Equal to det(A)
• Square of det(A)
• Zero

Correct Answer: Equal to det(A)

Q33. Which determinant property helps compute det(kA) for n×n matrix A?

• det(kA) = det(A) + k
• det(kA) = k^n det(A)
• det(kA) = k det(A) regardless of n
• det(kA) = det(A)^k

Correct Answer: det(kA) = k^n det(A)

Q34. If A is singular and B is invertible, then AB is:

• Always invertible
• Always singular
• Singular only if B is singular
• Non-singular if det(B) = 0

Correct Answer: Always singular

Q35. Which matrix must be non-singular?

• Zero matrix
• Identity matrix
• Matrix with two identical rows
• Matrix with determinant zero

Correct Answer: Identity matrix

Q36. For a block diagonal matrix, det of whole matrix equals:

• Sum of determinants of blocks
• Product of determinants of diagonal blocks
• Determinant of first block only
• Zero always

Correct Answer: Product of determinants of diagonal blocks

Q37. If det(A) ≠ 0, how many solutions does Ax = b have for each b?

• None
• Exactly one
• Infinitely many
• Two

Correct Answer: Exactly one

Q38. Which concept links singular matrices to linear dependence?

• Nullspace contains non-zero vectors
• Inverse is unique
• Determinant positive
• Columns form orthonormal basis

Correct Answer: Nullspace contains non-zero vectors

Q39. The cofactor expansion to compute determinant uses:

• Products of row sums
• Minors and cofactors recursively
• Only diagonal entries
• Only eigenvalues

Correct Answer: Minors and cofactors recursively

Q40. Which statement about triangular matrices is true?

• Triangular matrices are always singular
• Determinant equals product of diagonal entries
• Inverse never exists
• All off-diagonal entries determine determinant

Correct Answer: Determinant equals product of diagonal entries

Q41. If A has inverse, which of these is false?

• Columns of A are linearly independent
• Det(A) ≠ 0
• Nullspace of A contains non-zero vectors
• Ax = b has unique solution for each b

Correct Answer: Nullspace of A contains non-zero vectors

Q42. How does adding a multiple of one row to another affect determinant?

• Multiplies determinant by that multiple
• Changes determinant unpredictably
• Does not change determinant
• Sets determinant to zero

Correct Answer: Does not change determinant

Q43. Which is a quick way to see a matrix is non-singular numerically?

• Compute determinant and check it is close to zero within tolerance
• Check if any entry is zero
• Check if sum of columns equals zero
• Compute trace only

Correct Answer: Compute determinant and check it is close to zero within tolerance

Q44. If A is n×n and A^2 is singular, what can be said about A?

• A must be non-singular
• A must be singular
• A is orthogonal
• A has full rank

Correct Answer: A must be singular

Q45. Which of these matrices has determinant equal to product of eigenvalues (counting multiplicity)?

• Any square matrix
• Only diagonal matrices
• Only symmetric matrices
• Only orthogonal matrices

Correct Answer: Any square matrix

Q46. The inverse of a non-singular matrix can be used to:

• Find all eigenvectors
• Solve linear systems Ax = b efficiently
• Make A singular
• Change the rank of A

Correct Answer: Solve linear systems Ax = b efficiently

Q47. What does a determinant very close to zero indicate in practical computations?

• Matrix is well-conditioned
• Matrix may be ill-conditioned or nearly singular
• Matrix inverse is easy to compute
• Matrix columns are orthogonal

Correct Answer: Matrix may be ill-conditioned or nearly singular

Q48. In context of linear regression for drug response, singularity of X’X means:

• Unique least squares estimates exist
• Parameter estimates are not unique due to multicollinearity
• Model predictions are exact
• Design matrix is orthonormal

Correct Answer: Parameter estimates are not unique due to multicollinearity

Q49. Which of the following is a necessary and sufficient condition for A to be invertible?

• det(A) = 0
• The columns of A span R^n
• A has a zero row
• A has dependent columns

Correct Answer: The columns of A span R^n

Q50. For a continuous compartmental model linearized to Ax=b, what does singular A prevent?

• Having any solution
• Unique estimation of compartment transfer rates
• Model being linear
• Using matrix methods at all

Correct Answer: Unique estimation of compartment transfer rates

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