PRC-2 · Chapter 4
Linear Programming MCQs with Answers
60 multiple-choice questions on Linear Programming for PRC-2 Quantitative Analysis for Business. Try each one before revealing the answer and explanation.
Practise this chapter interactivelyQuestion 1
Which of the following best defines linear programming in a business context?
- A) A geometric method for estimating compounding future values.
- B) A statistical tool used to establish the correlation between two independent variables.
- C) A mathematical method for determining the best outcome, such as maximum profit, subject to limiting constraints.
- D) A formula used exclusively to solve quadratic equations.
Show answer & explanation
Answer: C) A mathematical method for determining the best outcome, such as maximum profit, subject to limiting constraints.
Linear programming is a mathematical optimization technique used to find the best possible outcome (like minimizing cost or maximizing profit) within a given mathematical model containing linear constraints.
Question 2
In a linear programming problem, what are the linear inequalities or restrictions placed on the variables formally called?
- A) Objective functions
- B) Constraints
- C) Feasible regions
- D) Progressions
Show answer & explanation
Answer: B) Constraints
Constraints represent the limiting factors in a business environment (like available labor hours or raw materials), which are mathematically expressed as linear inequalities or restrictions on the variables.
Question 3
Which specific method or theorem is widely used to find the optimal combination of values for x and y within a feasible region?
- A) The quadratic theorem
- B) The corner point theorem
- C) The progression formula
- D) The standard deviation method
Show answer & explanation
Answer: B) The corner point theorem
The corner point theorem states that the optimal solution (maximum or minimum value) of a linear programming problem will always occur at one of the corner points (vertices) of the feasible region.
Question 4
If a manufacturer has 4 workers, each working 8 hours a day, and it takes 2 hours of labor to produce one unit of product 'x', which inequality correctly represents the daily production constraint?
- A) 2x ≤ 32
- B) x ≥ 32
- C) 4x ≤ 8
- D) 2x = 12
Show answer & explanation
Answer: A) 2x ≤ 32
Total available labor hours per day are 4 workers * 8 hours = 32 hours. Since each unit takes 2 hours, the total labor time used (2x) must be less than or equal to the available time (32). Therefore, 2x ≤ 32.
Question 5
What is the term for a constraint in a linear programming model that does not impact the formation of the feasible region because another constraint is already more restrictive?
- A) Optimum constraint
- B) Feasible constraint
- C) Redundant constraint
- D) Negative constraint
Show answer & explanation
Answer: C) Redundant constraint
A redundant constraint is an inequality that, if removed from the system, would not change the shape or boundaries of the feasible region because other constraints already limit the area more strictly.
Question 6
Under what specific condition does the straight line representing a linear equation pass exactly through the origin (0,0)?
- A) When both x and y coefficients are negative.
- B) When the constant term is zero.
- C) When the inequality sign is strictly 'greater than'.
- D) When the line is perfectly vertical.
Show answer & explanation
Answer: B) When the constant term is zero.
An equation in the form of Ax + By = C will pass precisely through the origin (where x=0 and y=0) only if the constant term 'C' is equal to zero.
Question 7
Which of the following mathematical equations represents a straight line that passes through the origin?
- A) 2x + 5y = 10
- B) x - y = 4
- C) 3x - 4y = 0
- D) y = 2x + 1
Show answer & explanation
Answer: C) 3x - 4y = 0
Because the constant term is 0, plugging in x = 0 results in y = 0. Therefore, the line 3x - 4y = 0 intersects the origin (0,0).
Question 8
In a product mix decision, what does the inequality constraint x + y ≤ 150 signify?
- A) The difference between product x and y cannot exceed 150.
- B) The combined production volume of products x and y cannot exceed 150 units.
- C) The minimum production requirement for both products combined is 150.
- D) At least 150 units of product x must be produced.
Show answer & explanation
Answer: B) The combined production volume of products x and y cannot exceed 150 units.
The addition of x and y represents combined production, and the '≤' sign means 'less than or equal to'. Therefore, their total combined units cannot go over 150.
Question 9
For a strict production constraint given as x + y ≤ 80, what is the maximum possible value of x assuming production of y is halted entirely?
- A) 0
- B) 40
- C) 80
- D) 160
Show answer & explanation
Answer: C) 80
If the production of y is halted (y = 0), the inequality simplifies to x ≤ 80. Therefore, the maximum allowable value for x is 80.
Question 10
What is the primary business logic behind enforcing the non-negativity constraints x ≥ 0 and y ≥ 0 in linear programming models?
- A) To ensure that the company never takes a loan.
- B) Because physical quantities of products manufactured or sold cannot be negative.
- C) To force the objective function to always yield a positive profit.
- D) To place the feasible region exclusively in the third quadrant.
Show answer & explanation
Answer: B) Because physical quantities of products manufactured or sold cannot be negative.
In physical reality, a company cannot produce a negative number of items. The non-negativity constraint restricts the feasible region to the first quadrant to reflect this business reality.
Question 11
Given an objective function designed to minimize cost where C = 3x + 4y, and the corner points of the feasible region are (2, 5), (4, 1), and (0, 6). Which point yields the minimum cost?
- A) (2, 5)
- B) (4, 1)
- C) (0, 6)
- D) They all yield the same cost.
Show answer & explanation
Answer: B) (4, 1)
Testing the points: At (2, 5), C = 3(2) + 4(5) = 26. At (4, 1), C = 3(4) + 4(1) = 16. At (0, 6), C = 3(0) + 4(6) = 24. The minimum cost is 16, which occurs at point (4, 1).
Question 12
Identify the redundant constraint in the following system: x ≤ 5, y ≤ 6, and x ≤ 10.
- A) x ≤ 5
- B) y ≤ 6
- C) x ≤ 10
- D) There is no redundant constraint.
Show answer & explanation
Answer: C) x ≤ 10
Because the constraint x ≤ 5 already strictly limits the value of x to 5 or less, the constraint stating x must be ≤ 10 is redundant; it provides no additional restriction on the feasible region.
Question 13
Given the objective function to maximize profit P = 400x + 200y, and corner points of a feasible region at (0, 10), (8, 0), and (5, 5). What is the maximum profit achievable?
- A) Rs. 2,000
- B) Rs. 3,000
- C) Rs. 3,200
- D) Rs. 4,000
Show answer & explanation
Answer: C) Rs. 3,200
Evaluating the corner points: (0, 10) yields P = 2000. (8, 0) yields P = 400(8) = 3200. (5, 5) yields P = 400(5) + 200(5) = 2000 + 1000 = 3000. The maximum profit is Rs. 3,200 at (8, 0).
Question 14
Which of the following statements perfectly describes a feasible region?
- A) It is the single highest value calculated on a graph.
- B) It is the specific area on a graph that satisfies all the given linear inequalities and constraints simultaneously.
- C) It is the space where the objective function is automatically zero.
- D) It is an area defined purely by negative values.
Show answer & explanation
Answer: B) It is the specific area on a graph that satisfies all the given linear inequalities and constraints simultaneously.
The feasible region is defined mathematically and graphically as the continuous set of all possible points that satisfy every constraint imposed by the problem.
Question 15
A graphical boundary line is defined as x = 4. What does the inequality x ≥ 4 represent visually on the coordinate plane?
- A) The area exclusively to the left of the vertical line x = 4.
- B) The area exclusively below the horizontal line x = 4.
- C) The area extending to the right side of the vertical line x = 4.
- D) A single point located at coordinates (4, 0).
Show answer & explanation
Answer: C) The area extending to the right side of the vertical line x = 4.
The equation x = 4 represents a vertical line. The inequality x ≥ 4 includes the line itself and all the geometric space extending to the right, where x-values are greater than 4.
Question 16
What is the primary overarching purpose of utilizing linear programming in a business scenario?
- A) To precisely calculate the compound interest earned over time.
- B) To mathematically determine the best possible outcome, such as maximizing profit or minimizing cost, subject to limiting constraints.
- C) To predict the exact standard deviation of future sales.
- D) To establish a curvilinear relationship between two independent variables.
Show answer & explanation
Answer: B) To mathematically determine the best possible outcome, such as maximizing profit or minimizing cost, subject to limiting constraints.
Linear programming is a quantitative optimization technique. Its fundamental purpose is to find the optimal solution (like highest profit or lowest cost) for a mathematical model given a strict set of linear constraints.
Question 17
A factory produces items X and Y. Production is restricted by the inequalities x ≤ 20, y ≤ 30, and x ≤ 50. Which of these constraints is mathematically considered 'redundant'?
- A) x ≤ 20
- B) y ≤ 30
- C) x ≤ 50
- D) None of them are redundant
Show answer & explanation
Answer: C) x ≤ 50
A redundant constraint is one that does not actively limit the feasible region because another constraint is stricter. Since x is already strictly limited to 20 or less by the first constraint, the restriction x ≤ 50 has no further impact.
Question 18
In a linear programming graph, what specific term describes the geometric area that satisfies all the given linear inequalities simultaneously?
- A) The unbounded zone
- B) The objective region
- C) The feasible region
- D) The redundant area
Show answer & explanation
Answer: C) The feasible region
The feasible region is the overlapping shaded area on a coordinate graph where every single constraint (inequality) of the problem is met at the same time.
Question 19
The specific mathematical equation that a company seeks to maximize (such as total profit) or minimize (such as total costs) is formally called the:
- A) Constraint function
- B) Subjective function
- C) Feasible function
- D) Objective function
Show answer & explanation
Answer: D) Objective function
The objective function (often denoted as Z or P) is the primary linear equation in the model that represents the target goal the business is trying to optimize, like P = 10x + 15y.
Question 20
According to the fundamental theorems of linear programming, where does the optimal mathematical solution always occur within a defined, closed feasible region?
- A) Exactly in the geometric center of the region.
- B) At one of the outer corner points (vertices) of the region.
- C) At the origin point (0,0) exclusively.
- D) Along the x-axis boundary only.
Show answer & explanation
Answer: B) At one of the outer corner points (vertices) of the region.
The Corner Point Theorem states that if a linear programming problem has an optimal solution (maximum or minimum), it will always be found at one of the extreme corner points of the feasible region.
Question 21
Why are non-negativity constraints (x ≥ 0, y ≥ 0) standard requirements in production-based linear programming models?
- A) Because negative profits are illegal in accounting.
- B) To ensure the equations can be factored easily.
- C) Because physical quantities of products manufactured or resources consumed cannot be negative.
- D) To allow the line to cross into the third quadrant.
Show answer & explanation
Answer: C) Because physical quantities of products manufactured or resources consumed cannot be negative.
In real-world business scenarios, it is physically impossible to produce a negative number of tables or work negative hours. Non-negativity constraints restrict the graph to the first quadrant to reflect this reality.
Question 22
Given an objective function to maximize profit P = 50x + 40y, and the feasible corner points are (0,0), (10,0), (0,15), and (5,10). What is the maximum possible profit?
- A) 500
- B) 600
- C) 650
- D) 700
Show answer & explanation
Answer: C) 650
Test each corner point in the objective function: P(10,0) = 500. P(0,15) = 40(15) = 600. P(5,10) = 50(5) + 40(10) = 250 + 400 = 650. The maximum profit is 650.
Question 23
At what exact coordinate point do the linear boundary constraints x + y = 100 and 2x + y = 150 intersect?
- A) (25, 75)
- B) (50, 50)
- C) (75, 25)
- D) (100, 0)
Show answer & explanation
Answer: B) (50, 50)
Subtract the first equation from the second: (2x + y) - (x + y) = 150 - 100. This leaves x = 50. Substitute x = 50 into the first equation: 50 + y = 100, which means y = 50. The intersection is (50, 50).
Question 24
What is the mathematical slope of the boundary line defined by the production constraint 3x + 4y = 120?
- A) 3/4
- B) -3/4
- C) 4/3
- D) -4/3
Show answer & explanation
Answer: B) -3/4
To find the slope, rearrange the equation into the standard y = mx + c format. 4y = -3x + 120, which becomes y = (-3/4)x + 30. The slope 'm' is -3/4.
Question 25
For the raw material constraint 5x + 2y ≤ 200, what is the maximum number of product 'x' that can be produced if zero units of 'y' are produced?
- A) 40
- B) 50
- C) 100
- D) 200
Show answer & explanation
Answer: A) 40
To find the maximum x-value (the x-intercept), set y to 0. The inequality becomes 5x ≤ 200. Dividing both sides by 5 yields x ≤ 40.
Question 26
If the optimal solution of a linear program falls exactly on the intersection of two constraint lines, those specific constraints are mathematically referred to as:
- A) Redundant constraints
- B) Slack constraints
- C) Binding constraints
- D) Non-negativity constraints
Show answer & explanation
Answer: C) Binding constraints
Constraints that intersect to form the exact corner point where the optimal solution is found are called 'binding constraints' because they actively dictate and restrict the final optimal outcome.
Question 27
If a linear programming model contains mathematical constraints that severely contradict each other, resulting in absolutely no overlapping shaded area, the problem is considered:
- A) Unbounded
- B) Infeasible
- C) Redundant
- D) Optimal
Show answer & explanation
Answer: B) Infeasible
An 'infeasible' problem occurs when there is no possible combination of values that can satisfy all the given constraints simultaneously, meaning a feasible region simply does not exist.
Question 28
In a minimization problem, if the feasible region is open-ended and extends infinitely outward in the positive direction, the region is technically described as:
- A) Optimal
- B) Redundant
- C) Unbounded
- D) Infeasible
Show answer & explanation
Answer: C) Unbounded
When a feasible region is not fully closed by constraints and stretches endlessly toward infinity, it is called an 'unbounded' region. While it may still have a minimum point, it cannot have a maximum point.
Question 29
A workshop uses 3 machine hours to produce one unit of Alpha (x) and 4 machine hours for Beta (y). The total machine hours available per week is 120. Which inequality represents this?
- A) 3x + 4y ≥ 120
- B) 3x + 4y ≤ 120
- C) 4x + 3y ≤ 120
- D) x + y ≤ 120
Show answer & explanation
Answer: B) 3x + 4y ≤ 120
The total machine hours consumed is (3 * x) plus (4 * y). Because the factory cannot use more hours than it has available, this total must be less than or equal to (≤) the 120 available hours.
Question 30
A government contract requires a supplier to deliver a minimum combined total of 500 units of medical kits (x) and hygiene kits (y). Which inequality properly models this specific requirement?
- A) x + y ≤ 500
- B) x - y ≥ 500
- C) x + y = 500
- D) x + y ≥ 500
Show answer & explanation
Answer: D) x + y ≥ 500
The word 'minimum' implies that the supplier must provide 500 units or more. Therefore, the combined total of x and y must be 'greater than or equal to' 500, written as x + y ≥ 500.
Question 31
A loan of Rs. 1,000,000 is to be repaid over 10 years at a 12% annual interest rate. The repayment schedule follows an 'equated monthly installment' (EMI) structure. This form of payment is mathematically classified as:
- A) A perpetuity.
- B) An ordinary annuity.
- C) A simple interest growth curve.
- D) A deferred depreciation.
Show answer & explanation
Answer: B) An ordinary annuity.
An annuity is a sequence of equal payments made at regular intervals. Since EMIs occur at the end of each month for a fixed duration, they represent an ordinary annuity.
Question 32
What is the primary mathematical difference between 'Simple Interest' and 'Compound Interest'?
- A) Simple interest is always higher than compound interest.
- B) Simple interest is calculated only on the original principal, whereas compound interest is calculated on both the principal and the accumulated interest of previous periods.
- C) Compound interest does not apply to bank loans.
- D) Simple interest uses a logarithmic growth formula.
Show answer & explanation
Answer: B) Simple interest is calculated only on the original principal, whereas compound interest is calculated on both the principal and the accumulated interest of previous periods.
In simple interest, the interest amount remains constant because it applies only to the initial capital. In compound interest, the 'interest on interest' effect causes the debt or investment to grow exponentially.
Question 33
If a bank offers an annual nominal rate of 10% compounded 'quarterly', how many times (m) per year is interest added to the principal?
- A) 1
- B) 2
- C) 4
- D) 12
Show answer & explanation
Answer: C) 4
Compounded quarterly means the interest is calculated and added every three months, resulting in exactly 4 compounding periods within a single calendar year.
Question 34
What does the 'Effective Annual Rate' (EAR) represent in finance?
- A) The interest rate before any taxes are deducted.
- B) The annual interest rate including the impact of compounding within the year.
- C) The fixed rate set by the Central Bank.
- D) The rate applied to simple interest accounts only.
Show answer & explanation
Answer: B) The annual interest rate including the impact of compounding within the year.
The effective rate provides the 'true' annual cost or return. It accounts for the fact that compounding more frequently (e.g., monthly) results in a higher final yield than a simple nominal annual rate.
Question 35
In the formula for the Present Value of an Annuity, what is 'discounting'?
- A) Reducing the price of a product during a sale.
- B) The process of determining the current value of a future sum of money by removing the potential interest growth.
- C) Calculating the future value of a current investment.
- D) Increasing the principal by 10% annually.
Show answer & explanation
Answer: B) The process of determining the current value of a future sum of money by removing the potential interest growth.
Discounting is the inverse of compounding. It mathematically 'pulls' future cash flows back to today's value, reflecting the time value of money concept.
Question 36
An investment produces a perpetual stream of Rs. 50,000 every year forever. If the interest rate is 10%, what is the Present Value of this 'Perpetuity'?
- A) Rs. 50,000
- B) Rs. 500,000
- C) Rs. 5,000,000
- D) Zero
Show answer & explanation
Answer: B) Rs. 500,000
The Present Value of a perpetuity is calculated as (Annual Payment / Interest Rate). So, 50,000 / 0.10 = Rs. 500,000.
Question 37
If you invest Rs. 10,000 for 3 years at a simple interest rate of 8% per annum, what is the total interest earned?
- A) Rs. 800
- B) Rs. 2,400
- C) Rs. 12,400
- D) Rs. 2,597
Show answer & explanation
Answer: B) Rs. 2,400
Simple Interest = P × r × t. Here, 10,000 × 0.08 × 3 = Rs. 2,400.
Question 38
What happens to the Present Value of a future cash flow as the discount rate increases?
- A) It increases.
- B) It decreases.
- C) It remains constant.
- D) It becomes equal to the future value.
Show answer & explanation
Answer: B) It decreases.
The discount rate is in the denominator of the PV formula. As the rate (cost of capital) rises, the value today of receiving money in the future becomes lower.
Question 39
A 'Sinking Fund' is specifically designed to:
- A) Fund a company's daily operational losses.
- B) Accumulate a specific sum of money through periodic deposits to replace an asset or repay a debt in the future.
- C) Calculate the depreciation of a vehicle.
- D) Pay out dividends to shareholders immediately.
Show answer & explanation
Answer: B) Accumulate a specific sum of money through periodic deposits to replace an asset or repay a debt in the future.
A sinking fund is an application of annuities where regular installments are saved and invested to reach a targeted future financial goal.
Question 40
If interest is compounded 'continuously', which mathematical constant is utilized in the formula?
- A) Pi (π)
- B) Gamma (γ)
- C) Euler's Number (e)
- D) Delta (Δ)
Show answer & explanation
Answer: C) Euler's Number (e)
For continuous compounding, the growth is modeled using the exponential function A = Pe^(rt), where 'e' is approximately 2.71828.
Question 41
The 'Rule of 72' is a quick mental shortcut used to estimate:
- A) The total taxes on an investment.
- B) The time required for an investment to double at a given compound interest rate.
- C) The net present value of a bond.
- D) The maximum loan amount a person can borrow.
Show answer & explanation
Answer: B) The time required for an investment to double at a given compound interest rate.
By dividing 72 by the annual interest rate, you get an approximate number of years it takes for your money to double (e.g., at 6%, it takes ~12 years).
Question 42
In an 'Annuity Due', when are the periodic payments made?
- A) At the end of each period.
- B) At the beginning of each period.
- C) Only at the very end of the total duration.
- D) Randomly throughout the year.
Show answer & explanation
Answer: B) At the beginning of each period.
An ordinary annuity has payments at the end of the period. An 'Annuity Due' requires payment at the very start of each period (common for rent or insurance premiums).
Question 43
What is 'Amortization' in the context of a loan?
- A) The process of increasing the debt through penalties.
- B) The gradual reduction of a debt over time through regular payments of principal and interest.
- C) The immediate full repayment of a loan within 24 hours.
- D) Selling a loan to another bank.
Show answer & explanation
Answer: B) The gradual reduction of a debt over time through regular payments of principal and interest.
Amortization refers to spreading out the repayment of a loan into a series of fixed installments, ensuring the balance is zero at the end of the term.
Question 44
If the nominal rate is 12% and compounding is monthly, what is the 'periodic rate' used in calculations?
- A) 12%
- B) 6%
- C) 1%
- D) 0.12%
Show answer & explanation
Answer: C) 1%
The periodic rate is the annual nominal rate divided by the number of compounding periods in a year. 12% / 12 months = 1% per month.
Question 45
Which of the following is the standard formula for Compound Interest (A)?
- A) A = P(1 + r*t)
- B) A = P(1 + r)^n
- C) A = P / (1+r)
- D) A = P + r + t
Show answer & explanation
Answer: B) A = P(1 + r)^n
The standard compound interest formula reflects exponential growth, where the principal (P) is multiplied by (1 + rate) raised to the power of the number of periods (n).
Question 46
A matrix with the same number of rows as columns (e.g., 2x2 or 3x3) is known as a:
- A) Row Matrix
- B) Column Matrix
- C) Square Matrix
- D) Scalar Matrix
Show answer & explanation
Answer: C) Square Matrix
Square matrices are essential in business math as they are the only types that can have a determinant or an inverse.
Question 47
A square matrix where all diagonal elements are 1 and all other elements are 0 is called a/an:
- A) Null Matrix
- B) Identity Matrix (Unit Matrix)
- C) Scalar Matrix
- D) Transpose Matrix
Show answer & explanation
Answer: B) Identity Matrix (Unit Matrix)
The identity matrix (I) acts like the number '1' in matrix multiplication; multiplying any matrix by I leaves it unchanged.
Question 48
What is the condition for two matrices, A and B, to be 'conformable' for addition (A + B)?
- A) They must both be square matrices.
- B) They must have the same order (same number of rows and columns).
- C) The number of columns in A must equal the number of rows in B.
- D) Their determinants must both be non-zero.
Show answer & explanation
Answer: B) They must have the same order (same number of rows and columns).
To add or subtract matrices, you simply add/subtract corresponding elements, which requires both matrices to have identical dimensions.
Question 49
What is the condition for two matrices, A and B, to be conformable for *multiplication* (A × B)?
- A) They must have the same order.
- B) The number of columns in Matrix A must equal the number of rows in Matrix B.
- C) They must both be Identity matrices.
- D) Matrix B must be square.
Show answer & explanation
Answer: B) The number of columns in Matrix A must equal the number of rows in Matrix B.
Matrix multiplication involves 'row by column' operations, so the internal dimensions (cols of 1st and rows of 2nd) must match.
Question 50
If the determinant of a square matrix A is exactly zero (|A| = 0), the matrix is formally called:
- A) Non-singular
- B) Singular
- C) Invertible
- D) Diagonal
Show answer & explanation
Answer: B) Singular
A singular matrix does not have an inverse. This is critical in Cramer's Rule, as you cannot solve a system if the denominator (determinant) is zero.
Question 51
The process of swapping the rows and columns of a matrix A is called finding its:
- A) Inverse (A^-1)
- B) Determinant (|A|)
- C) Transpose (A^T)
- D) Adjoint
Show answer & explanation
Answer: C) Transpose (A^T)
If A is of order 2x3, then its transpose A^T will be of order 3x2.
Question 52
What is the determinant of the 2x2 matrix: [[5, 2], [3, 4]]?
- A) 11
- B) 14
- C) 20
- D) 26
Show answer & explanation
Answer: B) 14
Determinant = (ad - bc). Here, (5 × 4) - (2 × 3) = 20 - 6 = 14.
Question 53
To find the inverse of a 2x2 matrix, you must divide the 'Adjoint' of the matrix by its:
- A) Transpose
- B) Order
- C) Determinant
- D) Trace
Show answer & explanation
Answer: C) Determinant
Formula: A^-1 = (1 / |A|) × adj(A). This is why a matrix must be non-singular (|A| ≠ 0) to have an inverse.
Question 54
What is the product of a matrix A and its inverse A^-1?
- A) The Null Matrix (all zeros)
- B) The original matrix A
- C) The Identity Matrix (I)
- D) A scalar value of 1
Show answer & explanation
Answer: C) The Identity Matrix (I)
Just as 5 × (1/5) = 1 in regular math, a matrix multiplied by its inverse always results in the identity matrix.
Question 55
Cramer's Rule is a specific method used to solve:
- A) Matrix addition problems.
- B) Systems of simultaneous linear equations.
- C) Non-linear quadratic equations.
- D) Logarithmic sequences.
Show answer & explanation
Answer: B) Systems of simultaneous linear equations.
Cramer's Rule uses determinants to find the unique solution for variables in a system of equations.
Question 56
A matrix with only one single column is called a:
- A) Row Vector
- B) Column Vector
- C) Diagonal Matrix
- D) Square Matrix
Show answer & explanation
Answer: B) Column Vector
Column vectors are frequently used to represent variables or constants in matrix algebra.
Question 57
If you multiply a 2x3 matrix by a 3x4 matrix, the resulting matrix will have an order of:
- A) 2x3
- B) 3x4
- C) 2x4
- D) 3x3
Show answer & explanation
Answer: C) 2x4
The inner dimensions match (3 and 3), and the outer dimensions (2 and 4) determine the order of the product matrix.
Question 58
For a 2x2 matrix [[a, b], [c, d]], how do you find the 'Adjoint'?
- A) Multiply all elements by -1.
- B) Swap 'a' and 'd', and change the signs of 'b' and 'c'.
- C) Transpose the matrix twice.
- D) Take the square root of all elements.
Show answer & explanation
Answer: B) Swap 'a' and 'd', and change the signs of 'b' and 'c'.
This is a standard shortcut for finding the adjoint of a 2x2 matrix without calculating cofactors.
Question 59
Which of the following is true for matrix multiplication (where conformable)?
- A) AB is always equal to BA (Commutative law).
- B) AB is generally NOT equal to BA.
- C) (AB)^T is always equal to A^T × B^T.
- D) A × Null Matrix = Identity Matrix.
Show answer & explanation
Answer: B) AB is generally NOT equal to BA.
Matrix multiplication is non-commutative. Changing the order of the matrices usually changes the result (or makes it non-conformable).
Question 60
A matrix where every single element is zero (0) is called a:
- A) Identity Matrix
- B) Null Matrix (Zero Matrix)
- C) Scalar Matrix
- D) Inverse Matrix
Show answer & explanation
Answer: B) Null Matrix (Zero Matrix)
The null matrix acts like the number '0' in regular addition and multiplication.