Solution of system of linear equations (Matrix method) MCQs With Answer

Solution of system of linear equations (Matrix method) MCQs With Answer

Introduction: Mastering the solution of system of linear equations using matrix methods is essential for B.Pharm students tackling pharmaceutical calculations, formulation balancing, and kinetics modeling. This concise guide emphasizes key concepts — matrix representation (Ax = b), Gaussian elimination, Gauss-Jordan reduction, Cramer’s rule, determinants, inverse matrices, rank, and consistency — with practical pharmacy-related examples. Understanding these methods improves accuracy in dose calculations, mixing problems, and analysis of linear models in pharmacokinetics. The following questions reinforce procedural steps, interpretations of solutions (unique, infinite, none), and numerical techniques commonly used in drug formulation and laboratory calculations. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the correct matrix form of a linear system with coefficient matrix A, unknown vector x, and constants vector b?

• Ax = b
• A + x = b
• x = A + b
• Axb = 0

Correct Answer: Ax = b

Q2. Which method reduces a matrix to row echelon form to solve linear systems?

• Gaussian elimination
• Cramer’s rule
• Determinant expansion
• Matrix multiplication

Correct Answer: Gaussian elimination

Q3. What result indicates a unique solution for a square system Ax = b?

• det(A) ≠ 0
• det(A) = 0
• rank(A) < rank([A|b])
• rank(A) = 0

Correct Answer: det(A) ≠ 0

Q4. Which of the following operations is NOT an elementary row operation?

• Multiplying a row by a nonzero scalar
• Swapping two rows
• Adding a multiple of one row to another
• Multiplying two rows together

Correct Answer: Multiplying two rows together

Q5. Cramer’s rule is applicable when which condition is met?

• The coefficient matrix is square and has nonzero determinant
• The augmented matrix has higher rank than coefficient matrix
• The system is homogeneous only
• The matrix is singular

Correct Answer: The coefficient matrix is square and has nonzero determinant

Q6. The inverse of a 2×2 matrix [[a,b],[c,d]] exists and equals 1/(ad−bc) times which matrix?

• [[d, -b],[-c, a]]
• [[a, b],[c, d]]
• [[−d, b],[c, −a]]
• [[d, c],[b, a]]

Correct Answer: [[d, -b],[-c, a]]

Q7. What is the determinant of a 2×2 matrix [[p,q],[r,s]]?

• ps − qr
• pq + rs
• pr + qs
• p + q + r + s

Correct Answer: ps − qr

Q8. Which form does Gauss-Jordan elimination produce for a matrix?

• Reduced row echelon form (RREF)
• Upper triangular form only
• Lower triangular form only
• Diagonalization by eigenvectors

Correct Answer: Reduced row echelon form (RREF)

Q9. A homogeneous system Ax = 0 always has which solution?

• The trivial solution x = 0
• No solution
• A unique nonzero solution only
• Infinite solutions only if det(A) ≠ 0

Correct Answer: The trivial solution x = 0

Q10. If rank(A) = rank([A|b]) < number of unknowns, what is the nature of solutions?

• Infinitely many solutions
• Unique solution
• No solution
• Only trivial solution

Correct Answer: Infinitely many solutions

Q11. For pharmaceutical mixing problems modeled by linear equations, which step is first when using the matrix method?

• Set up coefficient matrix A, unknown vector x, and constants vector b
• Compute eigenvalues of A
• Differentiate the equations
• Apply Laplace transforms

Correct Answer: Set up coefficient matrix A, unknown vector x, and constants vector b

Q12. Which statement about a singular matrix is correct?

• It has determinant zero and is not invertible
• It always yields a unique solution for Ax = b
• It has full rank equal to its number of rows
• Its inverse exists

Correct Answer: It has determinant zero and is not invertible

Q13. Compute the determinant of matrix [[1,2,3],[0,1,4],[5,6,0]].

• 1
• 0
• −1
• 24

Correct Answer: 1

Q14. Solve the system x + y = 3 and x − y = 1 using matrix methods. What is x?

• 2
• 1
• 3
• 0

Correct Answer: 2

Q15. When using Gaussian elimination, what is a pivot?

• The first nonzero entry in a row used to eliminate variables below
• A determinant of a matrix
• A solution vector component
• The sum of row entries

Correct Answer: The first nonzero entry in a row used to eliminate variables below

Q16. Partial pivoting in Gaussian elimination is used primarily to:

• Improve numerical stability by swapping rows to get larger pivot entries
• Reduce the number of unknowns
• Compute eigenvectors faster
• Make the matrix symmetric

Correct Answer: Improve numerical stability by swapping rows to get larger pivot entries

Q17. The computational complexity of Gaussian elimination for an n×n system is approximately:

• O(n^3)
• O(n)
• O(n log n)
• O(2^n)

Correct Answer: O(n^3)

Q18. If A is invertible, the solution of Ax = b can be written as:

• x = A^(−1) b
• x = b A
• x = det(A) b
• x = 0

Correct Answer: x = A^(−1) b

Q19. Which is true about the rank of a matrix?

• It equals the maximum number of linearly independent rows or columns
• It is always equal to the number of rows
• It measures determinant magnitude
• It is undefined for rectangular matrices

Correct Answer: It equals the maximum number of linearly independent rows or columns

Q20. For a 3×3 system, Cramer’s rule requires computing how many determinants to find all unknowns?

• 4 determinants (one for denominator and one per unknown)
• 3 determinants only
• 1 determinant only
• 9 determinants

Correct Answer: 4 determinants (one for denominator and one per unknown)

Q21. When rank(A) < rank([A|b]), what does this imply?

• The system is inconsistent and has no solution
• The system has a unique solution
• The system is homogeneous
• The system has infinite solutions

Correct Answer: The system is inconsistent and has no solution

Q22. In pharmaceutical stoichiometry, linear systems can help find which of the following?

• Unknown quantities of components in a formulation
• Ionization constant of a drug
• Boiling point elevation constant
• Viscosity under shear stress

Correct Answer: Unknown quantities of components in a formulation

Q23. Which of the following best describes a consistent linear system?

• At least one solution exists
• No solution exists
• Exactly zero solutions exist
• It must be homogeneous

Correct Answer: At least one solution exists

Q24. In matrix terms, what does an augmented matrix represent?

• The coefficient matrix with the constants column appended
• The inverse of the coefficient matrix
• The transpose of the coefficient matrix
• A diagonalized version

Correct Answer: The coefficient matrix with the constants column appended

Q25. Which technique directly gives values of variables without computing the matrix inverse for a small square system?

• Cramer’s rule
• Eigenvalue decomposition
• Spectral clustering
• Matrix exponentiation

Correct Answer: Cramer’s rule

Q26. For the system represented by matrix A, if two rows are identical, what can be said about det(A)?

• det(A) = 0
• det(A) ≠ 0
• det(A) = 1
• det(A) is undefined

Correct Answer: det(A) = 0

Q27. Which property holds for determinant when a row is multiplied by scalar k?

• The determinant is multiplied by k
• The determinant remains unchanged
• The determinant becomes zero
• The determinant is squared

Correct Answer: The determinant is multiplied by k

Q28. A system with more variables than independent equations typically leads to:

• Infinitely many solutions if consistent
• A unique solution always
• No solution always
• Exactly one trivial solution only

Correct Answer: Infinitely many solutions if consistent

Q29. Which matrix equality indicates that two matrices are inverses?

• AB = I and BA = I
• A + B = I
• AB = 0
• A = B

Correct Answer: AB = I and BA = I

Q30. When solving Ax = b, why might one prefer LU decomposition?

• It allows efficient solution for multiple b vectors with same A
• It avoids row operations entirely
• It always gives integer solutions
• It computes determinants faster than Gaussian elimination

Correct Answer: It allows efficient solution for multiple b vectors with same A

Q31. In Gauss-Jordan elimination, after reaching RREF, the solution can be read directly because:

• Each pivot column corresponds to a leading variable with value shown
• The determinant is maximized
• The matrix is orthogonal
• All entries are zeros

Correct Answer: Each pivot column corresponds to a leading variable with value shown

Q32. In solving a 3×3 system by Cramer’s rule, D1 is computed by:

• Replacing the first column of A with b and taking its determinant
• Replacing the first row of A with b and taking its determinant
• Computing determinant of A and multiplying by x1
• Adding all rows of A

Correct Answer: Replacing the first column of A with b and taking its determinant

Q33. Which of the following is a disadvantage of Cramer’s rule in practical pharmacy calculations?

• It becomes computationally expensive for large systems
• It is inaccurate for 2×2 systems
• It cannot solve homogeneous systems
• It requires eigenvalues first

Correct Answer: It becomes computationally expensive for large systems

Q34. For augmented matrix [A|b], if a row becomes [0 0 … 0 | c] with c ≠ 0 after elimination, the system is:

• Inconsistent (no solution)
• Consistent with unique solution
• Consistent with infinite solutions
• Homogeneous

Correct Answer: Inconsistent (no solution)

Q35. Which of the following is true for a square matrix with linearly independent columns?

• It is invertible
• It has zero determinant
• It has repeated eigenvalues only
• It cannot be used to solve Ax = b

Correct Answer: It is invertible

Q36. In practice, when solving Ax = b numerically for pharmacokinetic models, which concern is important?

• Condition number of A affecting sensitivity to data errors
• Whether A is symmetric only
• Whether entries are prime numbers
• Whether b is orthogonal to A

Correct Answer: Condition number of A affecting sensitivity to data errors

Q37. Which approach can find a least-squares solution for an overdetermined system?

• Normal equations using A^T A x = A^T b
• Cramer’s rule directly
• Gauss-Jordan without transpose
• Compute determinant of augmented matrix

Correct Answer: Normal equations using A^T A x = A^T b

Q38. If two variables are dependent in a formulation model, what does that imply for the coefficient matrix?

• Columns are linearly dependent and rank is reduced
• Columns are orthogonal
• The matrix is diagonal
• The matrix must be invertible

Correct Answer: Columns are linearly dependent and rank is reduced

Q39. Which condition indicates a square matrix A has full rank?

• rank(A) = n for n×n matrix
• rank(A) = 0
• rank(A) < n
• rank(A) > n

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

Q40. When assembling a coefficient matrix for dissolution rate constants from linearized experimental equations, what units or scaling issue must be considered?

• Consistent units and scaling to avoid ill-conditioned matrices
• Only using integer coefficients
• Normalizing to determinant = 1
• Converting everything to percentages always

Correct Answer: Consistent units and scaling to avoid ill-conditioned matrices

Q41. Which statement about the transpose of a matrix A (denoted A^T) is true?

• Rows of A become columns in A^T
• Determinant of A^T is always zero
• A^T is the inverse of A
• A^T always equals A

Correct Answer: Rows of A become columns in A^T

Q42. In the context of drug formulation, using matrix methods helps to:

• Balance component proportions when multiple constraints exist
• Measure microbial contamination directly
• Replace laboratory testing entirely
• Predict patient adherence

Correct Answer: Balance component proportions when multiple constraints exist

Q43. A 3×3 coefficient matrix has determinant = 0. What is true about the solutions of Ax = b?

• There may be no solution or infinitely many solutions depending on b
• There is exactly one solution for every b
• Every b yields exactly three solutions
• There are always infinite solutions for any b

Correct Answer: There may be no solution or infinitely many solutions depending on b

Q44. Which of the following is a benefit of representing systems as matrices in pharmacology coursework?

• Compact representation and use of systematic elimination algorithms
• It avoids any computational error
• It guarantees integer solutions
• It eliminates the need for experimental data

Correct Answer: Compact representation and use of systematic elimination algorithms

Q45. If you scale a row by 3, then add it to another row during elimination, this is an example of which elementary row operation?

• Add a multiple of one row to another
• Swap two rows
• Multiply a row by a nonzero scalar only
• Transpose the matrix

Correct Answer: Add a multiple of one row to another

Q46. Which matrix decomposition is particularly useful for solving symmetric positive definite systems often arising in least-squares fitting of pharmacokinetic data?

• Cholesky decomposition
• QR decomposition only
• SVD with no advantages
• LU without pivoting

Correct Answer: Cholesky decomposition

Q47. In a 2×2 system, if A = [[2,1],[4,2]] and b = [3,6], what can you say about solutions?

• Infinite solutions (rows are proportional and b consistent)
• Unique solution
• No solution
• Determinant nonzero so unique

Correct Answer: Infinite solutions (rows are proportional and b consistent)

Q48. Which method finds the best-fit solution when the system Ax = b is inconsistent due to experimental errors?

• Least-squares solution via normal equations or QR decomposition
• Cramer’s rule
• Gauss-Jordan with exact arithmetic only
• Compute determinant of augmented matrix

Correct Answer: Least-squares solution via normal equations or QR decomposition

Q49. The Moore-Penrose pseudoinverse is used primarily to:

• Find least-squares solutions for non-square or rank-deficient systems
• Compute exact Cramer’s-rule solutions
• Invert only diagonal matrices
• Diagonalize matrices only

Correct Answer: Find least-squares solutions for non-square or rank-deficient systems

Q50. When preparing MCQ-based practice for B.Pharm students on matrix methods, which focus will most improve applied competence?

• Emphasizing setup from real pharmacy problems, elimination steps, interpretation of results, and numerical stability
• Memorizing determinant formulas without context
• Only practicing textbook examples with no application
• Avoiding numerical methods and focusing on theory only

Correct Answer: Emphasizing setup from real pharmacy problems, elimination steps, interpretation of results, and numerical stability

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