UTK Math 123 Final True/False Guide

If a system of linear equations is inconsistent, then it has infinitely many solutions.

False; "inconsistent" means there are no solutions due to parallel lines

No LP problem with an unbounded feasible region has a solution.

False; a minimization problem can.

If defined, a column times a row is never a 1 × 1 matrix.

False; if both matrices are 1 x 1 the result would be a 1 x 1.

When we row reduce a matrix, we must always turn each pivot into a 1 before clearing its column, or else errors will result.

False; the order in which it is reduced does not matter.

If A is a 2 × 3 matrix and B is a 3 × 2 matrix, then the sum A + B is defined.

False; the sum can only be found for square matrices with the same order.

Some row reduced matrices have a 2 in the top left-hand corner.

False; the value must be a 1 or 0 (if matrix is 1x1)

[ 1 2 ] [ 2 1 ] is singular.

False; this matrix does have an inverse

The ij entry of the product AB is obtained by multiplying the ith column of A by the jth row of B.

False; use the row of A and the column of B

If an LP problem has a solution at all, it will have a solution at some corner of the feasible region.

True

If the value of a strictly determined game is positive, it favors the row player.

True

If, at any stage of an iteration of the simplex method, it is not possible to compute the ratios (division by zero) or the ratios are negative, then the standard linear programming problem has no solution.

True

In a standard maximization problem, each constraint inequality must be written so that it is less than or equal to a nonnegative number.

True

In a standard minimization problem, each constraint inequality must be written so that it is greater than or equal to any real number.

True

In a strictly determined game, the value of the game is given by the saddle point of the game.

True

In a zero-sum game, the payments made by the players at the end of each play add up to zero.

True

In the final tableau of a simplex method problem, if the problem has a solution, the last column will contain no negative numbers above the bottom row.

True

In the row-reduced form of a matrix, the first nonzero entry in each row is a 1.

True

The dual of a standard minimization problem with nonnegative objective function coefficients is a standard maximization problem.

True

The feasible region of a LP problem with two unknowns may be bounded or unbounded.

True

The following LP problem has an unbounded feasible region: Minimize c = x − y subject to 4x − 3y ≥ 0 3x − 4y ≤ 0 x ≥ 0, y ≥ 0

True

The following LP problem optimal solution (0, 0): Minimize c = x − y subject to 4x − 3y ≤ 0 3x − 4y ≥ 0 x ≥ 0, y ≥ 0

True

The following is a standard maximum problem: Maximize p = −2x − 3y − z subject to 4x − 3y + z ≤ 3 −x − y − z ≤ 10 2x + y − z ≤ 10 x ≥ 0, y ≥ 0, z ≥ 0

True

In a feasible basic solution all the variables (with the possible exception of the objective) are nonnegative.

True; x ≥ 0, y ≥ 0, z ≥ 0

If A is a 2 × 3 matrix and B is a 3 × 2 matrix, then the product AB is defined.

True; A must have the same number of columns as B has rows to successfully multiply.

If a feasible region is empty, then it is bounded.

True; If the feasible region is empty, then there is no maximum or minimum values. An empty region results when there are no points that satisfy all of the constraints.

Some row reduced matrices have a 0 in the top left-hand corner.

True; a 1x1 matrix justifies this claim [0]

If the row-reduced form of a square matrix contains a row of zeros, then the matrix is singular.

True; a row of all zeros yields the matrix to not have an inverse

The ij entry of the product AB is obtained by multiplying the ith row of A by the jth column of B.

True; for example if the desired position is 34, one would multiply (row 3 of A) by (column 4 of B)

In the simplex method, a basic solution can assign the value zero to some basic variables.

True; for example, using (x,y,z): P(1,0,2)=42 where y=0

Some LP problems have more than one solution.

True; if objective function is parallel to an inequality that contains two corners of the feasible, there is more than one solution.

If two rows of a square matrix are equal, then the matrix is singular.

True; if the rows are equal, once reduced that will result in an all zero row which does not have an inverse.

If the graphs of two linear equations are not parallel, then there is a unique solution to the system.

True; if they are not parallel, the functions must intersect at some point

If the row-reduced form of a matrix is the identity, then the matrix is invertible.

True; it can also be classified as non-singular

If two linear equations have the same graph, then the associated system has infinitely many solutions.

True; the parallel lines are overlapping to for infinitely many solutions.

If two of the equations in a system of linear equations are inconsistent, then the whole system is inconsistent.

True; this is because there is not a point where all equations intersect resulting in no solution.

A system of three equations in two unknowns cannot have a solution.

False

According to the principles of game theory, your opponent can always anticipate your move.

False

Every LP problem in two unknowns has optimal solutions.

False

If AB = 0, then either A or B is a zero matrix.

False

If a game has expected value 2, then the row player will gain two points on every play assuming both players use their optimal mixed strategies.

False

In the simplex method, a basic solution assigns the value zero to all active variables.

False

Some strictly determined games do not have saddle points.

False

The dual of a standard minimization problem must be a standard maximization problem.

False

The following LP problem has an empty feasible region: Minimize c = x − y subject to 4x − 3y ≤ 0 3x − 4y ≥ 0 x ≥ 0, y ≥ 0

False

The following LP problem has an unbounded feasible region: Minimize c = x − y subject to 4x − 3y ≤ 0 3x − 4y ≥ 0 x ≥ 0, y ≥ 0

False

The following LP problem has an unbounded feasible region: Minimize c = x − y subject to 4x − 3y ≤ 0 x + y ≤ 10 x ≥ 0, y ≥ 0

False

The following is a standard maximum problem: Maximize p = x − y − 3z subject to 4x − 3y − z ≤ −3 x + y + z ≤ 10 2x + y − z ≤ 10 x ≥ 0, y ≥ 0, z ≥ 0

False

The payoff matrix is always a square matrix.

False

The system x = y; y = z, x = z is inconsistent.

False

The transpose of a 5 × 6 matrix has six columns and five rows.

False

[ 1 1 ] [ 3 3 ] is invertible.

False

If a system of linear equations has infinitely many solutions, then it may be inconsistent.

False: inconsistent is defined as having no solutions due to parallel lines.

Every system of three linear equations in three unknowns has at least one solution.

False: one example would be if two of the equations are parallel, there would be no solution due to no intersection of all three equations.

In the row-reduced form of a matrix, the first non zero entry in each row must be a 1 on a diagonal axis.

False: while the first nonzero value in each row must be 1, it does not have to be on a diagonal axis.

It is never true that A + B, A − B and AB are all defined.

False;

If quantity x is twice quantity y, then 2x − y = 0.

False; 2x - y = 0 @ (2,1) 2(2) - 1 = 3 3 ≠ 0

If a system of linear equations is represented by AX = B and A is invertible, then the system has infinitely many solutions.

False; [a b] [x] = [b1] [c d] [y] [b2]

If a system of linear equations is represented by AX = B and A is not invertible, then the system has no solution.

False; [a b] [x] = [b1] [c d] [y] [b2]

The system x + y + z = 3; x = y; y = z, y = 1 is inconsistent.

False; 1+1+1=3

Some LP problems have exactly two solutions.

False; LP solutions occur at corners. Results are either one solution, infinitely many (objective parallel to max function), or no solution (unbounded toward x and/or y).

The optimal value attained by the objective function for the primal problem may be different from that attained by the dual problem.

False; Say the primal was a maximization problem. The max for the primal problem will equal the min of the dual problem.

A row reduced matrix always has a 1 in the second column of the second row.

False; a zero matrix is reduced but does not have a 1 in the {22} location.

If the row-reduced form of a matrix is the identity, then the matrix is singular.

False; an identity matrix (square matrix with ones on the main diagonal and zeros elsewhere) is easily invertible.

If the graphs of two linear equations are not parallel, then there may be no solution to the system.

False; an intersection will be present if the two linear equations are not parallel

If the graphs of two linear equations are parallel and distinct, then there is a unique solution to the system.

False; due to the two lines never coming in contact with each other, there is no solution.

Your opponent's knowledge that you are using some specific non-optimal strategy does not benefit him in the least.

False; if they know which specific non-optimal strategy you are using, they can alter their current strategy.

The solution set of ax + by < c is either a left half-plane or a lower half-plane.

False; if x=10000 and y = 10000 while c=1, it would be either a right half-plane or an upper half-plane.

The graph of a linear inequality consists of a line and some points on both sides of the line.

False; inequality entails that the values are on one side of the function.

The graph of a linear inequality consists of a line and only some of the points on one side of the line.

False; it consists of a line and ALL points on one side of the line.

The solution set of the inequality ax + by + c ≤ 0 is either a left half-plane or a lower half-plane.

False; it is either a right half-plane or upper half-plane. This is because if the left side of the inequality is greater than 0, above the line would be shaded.

In a standard minimization problem, each constraint inequality must be written so that it is greater than or equal to a nonnegative number.

False; it must be greater than or equal to any real number

Every minimization problem can be converted into a standard maximization problem.

False; not every min problem can become a STANDARD max problem

If A and B are 2 × 2 matrices such that AB = 0, then BA = 0.

False; the commutative property off multiplication does not apply to matrix multiplication.

An optimal strategy is one that maximizes one's maximum potential gain.

False; the goal is to minimize losses.

If the row reduced form of a matrix has more than one non-zero entry in any row, then the corresponding system of linear equations has infinitely many solutions.

False; the linear equations may be inconsistent

If both players use their optimal pure strategies, then the game is fair.

False; value of the game must be zero to be fair.

"Not invertible" is the same thing as "singular."

True

A negative payoff indicates a loss to the row player.

True

A non singular matrix is a square matrix that is invertible (has an inverse).

True

A singular matrix is a square matrix that is not invertible (does not have an inverse).

True

A sinking fund is an account from which money is being deposited.

True

An "inconsistent system" is when there is no solution because the lines are parallel.

True

An "unknown" is synonymous with a variable (x,y,z,etc)

True

An annuity is an account from which money is being withdrawn

True

An optimal strategy is one that minimizes the maximum damage the opponent can cause.

True

Choosing the pivot column by requiring that it be the column associated with the most negative entry to the left of the vertical line in the last row of the simplex tableau ensures that the iteration will result in the greatest increase, or, at worst, no decrease in the objective function.

True

Choosing the pivot row by requiring that the ratio associated with that row be the smallest non-negative number insures that the iteration will not take us from a feasible point to a non-feasible point.

True

Different saddle points in the same payoff matrix always have the same payoff.

True

Every minimization problem can be converted into a maximization problem.

True

If A and B are diagonal 3 × 3 matrices, then AB = BA.

True

The following is a standard minimization problem: Minimize c = −x − y − 3z subject to 4x − 3y − z ≥ 3 x + y + z ≥ 10 2x + y − z ≥ 0 x ≥ 0, y ≥ 0, z ≥ 0

True

The simplex method can be used to solve all LP problems that have solutions.

True

The system of equations ax + by = 0; cx + dy = 0 has at least one solution regardless of the values of a, b, c, d.

True

Your opponent's knowledge that you are using your optimal strategy does not benefit him in the least.

True

[ 1 2 ] [ 1 1 ] is invertible.

True

If quantity x is twice quantity y, then x − 2y = 0.

True; x - 2y = 0 @ (2,1). 2 - 2(1) =0

If a system of linear equations has two solutions, then it has infinitely many solutions.

True; this means that the objective function is parallel to the linear equation that contains the two solutions.

You are mixing x grams of ingredient A and y grams of ingredient B. Choose the equation or inequality that models the requirement: There should be at least 3 times the grams of ingredient A than ingredient B.

x - 3y ≥ 0

You are mixing x grams of ingredient A and y grams of ingredient B. Choose the equation or inequality that models the requirement: There should be no more grams of ingredient A than ingredient B.

x - y ≤ 0

You are mixing x grams of ingredient A and y grams of ingredient B. Choose the equation or inequality that models the requirement: There should be at least 3 more grams of ingredient A than ingredient B.

x - y ≥ 3