Inverse of a matrix MCQs With Answer

Inverse of a matrix MCQs With Answer is an essential topic in matrix algebra for B. Pharm students studying pharmacokinetics, compartment models and quantitative methods. This concise introduction explains how the matrix inverse helps solve linear systems, estimate parameters, and model drug distribution. Key ideas include determinant non‑zero tests, adjoint and Gaussian elimination methods, numerical stability, pseudoinverse for non‑square systems, and practical applications in drug concentration modeling and regression. Mastering these concepts improves problem solving in dosing models and laboratory data analysis. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the defining property of the inverse of a square matrix A?

• AA = I
• A + A^-1 = I
• AA^-1 = A^-1A = I
• A^-1 = 0

Correct Answer: AA^-1 = A^-1A = I

Q2. A square matrix is invertible if and only if which condition holds?

• Its determinant is zero
• Its determinant is non‑zero
• It has at least one zero row
• It is symmetric

Correct Answer: Its determinant is non‑zero

Q3. What is the inverse of a 2×2 matrix [[a,b],[c,d]] when ad − bc ≠ 0?

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

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

Q4. Which statement is true for the inverse of the identity matrix I of any size?

• I^-1 = 0
• I^-1 = I
• I^-1 = -I
• I^-1 does not exist

Correct Answer: I^-1 = I

Q5. If A and B are invertible square matrices, what is (AB)^-1?

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

Correct Answer: B^-1 A^-1

Q6. Which method is commonly used to compute an inverse numerically by augmenting with the identity?

• LU factorization
• Gaussian elimination on [A | I]
• Eigen decomposition only
• Determinant expansion

Correct Answer: Gaussian elimination on [A | I]

Q7. For a diagonal matrix with nonzero diagonal entries, what is its inverse?

• A matrix of zeros
• A diagonal matrix of reciprocals of the diagonal entries
• Transpose of the diagonal matrix
• Negative of the diagonal matrix

Correct Answer: A diagonal matrix of reciprocals of the diagonal entries

Q8. If A is orthogonal (A^T A = I), what is A^-1?

• A^T
• -A
• Determinant of A
• A^2

Correct Answer: A^T

Q9. The determinant of A^-1 equals which of the following?

• det(A)
• -det(A)
• 1/det(A)
• 0

Correct Answer: 1/det(A)

Q10. If A has eigenvalues λ1, λ2, …, λn (nonzero), what are the eigenvalues of A^-1?

• λ1^2, λ2^2, …
• 1/λ1, 1/λ2, …
• -λ1, -λ2, …
• Same as A

Correct Answer: 1/λ1, 1/λ2, …

Q11. Which property holds for transpose and inverse?

• (A^T)^-1 = (A^-1)^T
• (A^T)^-1 = A
• (A^T)^-1 = -A^-1
• Transpose and inverse are unrelated

Correct Answer: (A^T)^-1 = (A^-1)^T

Q12. In pharmacokinetic compartment models represented by linear ODEs, inversion of which matrix helps compute steady state solutions?

• Coefficient matrix of the linear system
• Identity matrix only
• Zero matrix
• Transpose of the time vector

Correct Answer: Coefficient matrix of the linear system

Q13. Which decomposition is commonly used to compute inverse more stably than direct inversion for large systems?

• QR or LU decomposition
• Direct determinant inversion
• Counting method
• None — direct inverse is always best

Correct Answer: QR or LU decomposition

Q14. If A is singular, which statement is true?

• A^-1 exists and is unique
• A^-1 does not exist
• All entries of A must be zero
• A has full rank

Correct Answer: A^-1 does not exist

Q15. What is the Moore‑Penrose pseudoinverse used for?

• Inverting non‑square or singular matrices to get least squares solutions
• Computing only for orthogonal matrices
• Replacing determinants in 2×2 matrices
• Finding eigenvectors only

Correct Answer: Inverting non‑square or singular matrices to get least squares solutions

Q16. Which condition number behaviour indicates numerical instability when inverting A?

• Condition number ≈ 1
• Condition number very large
• Condition number negative
• Condition number equal to zero

Correct Answer: Condition number very large

Q17. What is the inverse of a scalar multiple kA (k ≠ 0)?

• (kA)^-1 = k A^-1
• (kA)^-1 = (1/k) A^-1
• (kA)^-1 = A^-1 / k^2
• (kA)^-1 does not exist

Correct Answer: (1/k) A^-1

Q18. For triangular matrices, when is the inverse guaranteed to exist?

• When all diagonal entries are non‑zero
• When any one diagonal entry is zero
• Only if symmetric
• Only if diagonal entries sum to zero

Correct Answer: When all diagonal entries are non‑zero

Q19. Which of the following equals (A^-1)^-1?

• I
• -A
• A
• A^T

Correct Answer: A

Q20. In solving Ax = b, why is computing A^-1 and then x = A^-1 b sometimes discouraged?

• It is always faster
• It can be numerically unstable and inefficient compared to solving via factorization
• It gives wrong solutions
• It only works for diagonal matrices

Correct Answer: It can be numerically unstable and inefficient compared to solving via factorization

Q21. If AB = I for square matrices A and B, what can be concluded?

• B is a left inverse but not necessarily right inverse
• Both A and B are invertible and B = A^-1
• Only A is invertible
• A is singular

Correct Answer: Both A and B are invertible and B = A^-1

Q22. Which matrix inverse property is false?

• (A + B)^-1 = A^-1 + B^-1
• (AB)^-1 = B^-1 A^-1
• (A^T)^-1 = (A^-1)^T
• (A^-1)^-1 = A

Correct Answer: (A + B)^-1 = A^-1 + B^-1

Q23. The inverse of a permutation matrix is which of the following?

• Its transpose
• Negative permutation
• Zero matrix
• Diagonal matrix

Correct Answer: Its transpose

Q24. Which statement about the inverse and trace is true?

• trace(A^-1) = 1/trace(A)
• trace(A^-1) equals determinant of A
• No simple relation exists between trace(A) and trace(A^-1) in general
• trace(A^-1) = -trace(A)

Correct Answer: No simple relation exists between trace(A) and trace(A^-1) in general

Q25. In ordinary least squares regression, (X^T X)^-1 X^T y computes:

• Predicted y only
• Regression coefficients (β̂)
• Residuals only
• Determinant of X

Correct Answer: Regression coefficients (β̂)

Q26. Which approach helps when A is nearly singular to obtain stable parameter estimates?

• Tikhonov regularization (ridge), adding λI
• Directly inverting A with high precision only
• Removing all nonzero rows
• Transposing A repeatedly

Correct Answer: Tikhonov regularization (ridge), adding λI

Q27. For a 2×2 matrix [[2,5],[1,3]], which of the following is its inverse?

• [[3, -5],[-1, 2]]
• 1/(6−5) * [[3, -5],[-1, 2]]
• 1/(2−15) * [[3, -5],[-1, 2]]
• [[2,5],[1,3]]

Correct Answer: 1/(6−5) * [[3, -5],[-1, 2]]

Q28. What is the inverse of a 3×3 block diagonal matrix with invertible diagonal blocks?

• Block diagonal matrix of the inverses of each block
• Full dense inverse unrelated to blocks
• Zero matrix
• Transpose of the original block diagonal

Correct Answer: Block diagonal matrix of the inverses of each block

Q29. If rank(A) < n for an n×n matrix, what does this imply about A^-1?

• A^-1 exists
• A^-1 does not exist
• A is orthogonal
• A is diagonalizable

Correct Answer: A^-1 does not exist

Q30. When using the adjugate (classical adjoint) method, the inverse equals which expression?

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

Correct Answer: adj(A) / det(A)

Q31. Which matrix type always has an inverse if its diagonal entries are all positive and symmetric?

• Skew‑symmetric
• Symmetric positive definite
• Singular symmetric
• Nilpotent

Correct Answer: Symmetric positive definite

Q32. In discrete compartmental pharmacokinetics, solving steady state often requires inversion of which matrix built from rate constants?

• Time vector
• Rate constant coefficient matrix
• Identity matrix only
• Observation vector

Correct Answer: Rate constant coefficient matrix

Q33. The computational complexity of naive matrix inversion using Gaussian elimination is approximately:

• O(n)
• O(n^2)
• O(n^3)
• O(log n)

Correct Answer: O(n^3)

Q34. Which is true about the inverse of a symmetric matrix A (when A is invertible)?

• A^-1 is always skew‑symmetric
• A^-1 is also symmetric
• A^-1 is always diagonal
• A^-1 does not exist

Correct Answer: A^-1 is also symmetric

Q35. Which of the following is a correct application of matrix inverse in B.Pharm data analysis?

• Estimating regression coefficients in linear models
• Measuring pH directly
• Counting bacterial colonies by hand
• Labeling chemical structures

Correct Answer: Estimating regression coefficients in linear models

Q36. If A is invertible and B = A^-1, what is AB?

• Zero matrix
• Identity matrix
• A itself
• B itself

Correct Answer: Identity matrix

Q37. In numeric computations, which factorization is often used to solve Ax = b without explicitly computing A^-1?

• LU factorization
• Counting inverse entries
• Direct determinant inversion
• Adjugate only

Correct Answer: LU factorization

Q38. Which of these matrices always has an inverse?

• Zero matrix
• Identity matrix
• Matrix with a zero row
• Matrix with determinant zero

Correct Answer: Identity matrix

Q39. For non‑square m×n matrices with m > n, which inverse concept yields least squares solutions?

• Right inverse only
• Left inverse only
• Moore‑Penrose pseudoinverse
• Transpose inverse

Correct Answer: Moore‑Penrose pseudoinverse

Q40. Which statement about the product of determinants is correct?

• det(AB) = det(A) + det(B)
• det(AB) = det(A) det(B)
• det(AB) = det(A) / det(B)
• det(AB) unrelated to det(A) and det(B)

Correct Answer: det(AB) = det(A) det(B)

Q41. If A is invertible, which is true about nullspace of A?

• Nullspace contains nonzero vectors
• Nullspace is {0} only
• Nullspace equals entire space
• Nullspace is undefined

Correct Answer: Nullspace is {0} only

Q42. Which of the following best describes numerical regularization to handle near singular matrices?

• Adding a small multiple of the identity to A
• Removing diagonal entries
• Setting determinant to zero
• Multiplying A by zero

Correct Answer: Adding a small multiple of the identity to A

Q43. Which of these is a necessary and sufficient condition for invertibility of a square matrix A?

• Columns of A are linearly independent
• A has at least one zero column
• A is symmetric
• trace(A) = 0

Correct Answer: Columns of A are linearly independent

Q44. If A is unitary (complex analogue of orthogonal), what is A^-1?

• A conjugate transpose (A*)
• -A
• Inverse does not exist
• A itself

Correct Answer: A conjugate transpose (A*)

Q45. Which operation verifies that a computed matrix B is the inverse of A?

• Compute AB and check if equals A
• Compute AB and check if equals I
• Compute A + B and check if equals I
• Compute det(A) + det(B)

Correct Answer: Compute AB and check if equals I

Q46. Inverting a matrix with very small determinant often leads to which numerical issue?

• High precision improvements
• Large rounding errors and instability
• Faster computations
• Guaranteed exact result

Correct Answer: Large rounding errors and instability

Q47. Which is true about inverse of a symmetric positive definite matrix used in covariance estimation?

• Its inverse is not meaningful
• Its inverse is also symmetric positive definite
• Its inverse is skew‑symmetric
• Its inverse must be diagonal

Correct Answer: Its inverse is also symmetric positive definite

Q48. For a 2×2 matrix [[4,7],[2,6]], which option gives the correct inverse?

• 1/(24−14) * [[6, -7],[-2, 4]]
• 1/(4−14) * [[6, -7],[-2, 4]]
• 1/(24+14) * [[6, 7],[2, 4]]
• [[4,7],[2,6]]

Correct Answer: 1/(24−14) * [[6, -7],[-2, 4]]

Q49. Which statement correctly connects rank and invertibility for n×n matrices?

• A is invertible iff rank(A) = n
• A is invertible iff rank(A) = 0
• A invertible implies rank(A) < n
• Rank unrelated to invertibility

Correct Answer: A is invertible iff rank(A) = n

Q50. Which approach is preferred in practice to solve multiple systems Ax = b_k (different b_k) efficiently?

• Compute A^-1 using adjugate for each b_k
• Compute one LU or factorization of A then solve for each b_k
• Compute determinant for each b_k
• Use random guesses for x

Correct Answer: Compute one LU or factorization of A then solve for each b_k

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