Flashcards: Linear Equations and Matrix Algebra — 94 cards

What is an algebraic expression?

A mathematical statement linking numerical quantities by operations.

What does an equation indicate about two algebraic expressions?

They are equal.

What characterizes a linear equation's variables?

They have degree one.

What types of terms does a linear equation contain?

Only constants or constants times one variable to the first power.

What does the slope of a line measure?

The change in Y relative to a one-unit change in X.

What else does the slope measure about a line?

The line's steepness.

What is the general form of a linear equation?

$Y=mx+b$

In $Y=mx+b$, what does $m$ represent?

The slope.

Which forms can develop a line equation?

Slope-intercept, slope-point, and two-point forms.

What is the slope-intercept form of a line?

$Y=mX+b$ where $m$ is slope and $b$ is Y-intercept.

What is the slope-point form of a line through $(X_1,Y_1)$?

$Y - Y_1 = m(X - X_1)$ where $m$ is slope.

How is slope $m$ calculated from points $(X_1,Y_1)$ and $(X_2,Y_2)$?

$m=\frac{Y_2 - Y_1}{X_2 - X_1}$ when $X_1 \ne X_2$.

When is a line horizontal and what is its equation?

When points share the same Y-coordinate; equation is $Y=\text{constant}$.

When is a line vertical and what is its equation?

When points share the same X-coordinate; equation is $X=\text{constant}$.

When are two non-vertical lines parallel?

When they have the same slope.

When are two non-vertical lines perpendicular?

When the product of their slopes is $-1$.

What is the formula for total cost in business cost models?

$TC=VC+FC$

How is total revenue calculated with price $P$ and quantity $Q$?

Total revenue is $TR=PQ$.

How is profit $\Pi$ calculated from total revenue and total cost?

Profit is $\Pi=TR-TC$.

In a linear cost model, what equals the unit variable cost?

Marginal cost equals unit variable cost.

How does average fixed cost behave as quantity increases in a linear cost model?

Average fixed cost decreases as quantity increases.

What defines the break-even point in business?

It is where total revenue equals total cost and profit is zero.

What is the formula for break-even quantity $Q_e$ in manufacturing?

$Q_e=\frac{FC}{P-VC}$

How does increasing selling price affect break-even quantity?

Increasing selling price decreases break-even quantity.

What is a matrix in terms of its arrangement?

A rectangular array of real numbers arranged in m rows and n columns.

Where is element $a_{ij}$ located in a matrix?

In row i and column j of the matrix.

What does the dimension $m \times n$ of a matrix represent?

It means the matrix has m rows and n columns.

Which main types of matrices are commonly presented?

Vector, square, zero, identity, and scalar matrices.

What defines an identity matrix?

A square matrix with ones on the primary diagonal and zeros elsewhere.

What happens when you multiply a conformable matrix by an identity matrix?

The matrix remains unchanged after multiplication.

When are two matrices conformable for addition or subtraction?

Only when they have the same dimensions.

What does multiplying a matrix by a scalar do to its elements and dimension?

It multiplies every element by the scalar and keeps the same dimension.

What property does scalar multiplication of matrices satisfy with a real scalar X?

Scalar multiplication commutes: X A = A X.

How is scalar multiplication distributive over scalar and matrix addition?

(X + Y)A = XA + YA and X(A + B) = XA + XB.

What is the associative property of scalar multiplication?

X(YA) = XY(A).

When are two matrices conformable for multiplication?

When the first matrix's columns equal the second matrix's rows.

What is the dimension of the product AB if A is n × m and B is m × p?

AB has dimension n × p.

Which properties does matrix multiplication have and which does it generally lack?

It is associative and distributive but not generally commutative.

Can AB be zero if neither A nor B is zero?

Yes, AB may be zero even if neither A nor B is zero.

Does AB = AC imply B = C for matrices?

No, AB = AC does not generally imply B = C.

What condition defines the inverse of a square matrix A?

AA⁻¹ = A⁻¹A = I

What distinguishes an invertible matrix from a singular matrix?

An invertible matrix has an inverse; a singular matrix does not.

Which matrices can have inverses?

Only square matrices can have inverses.

Is the inverse of a matrix unique when it exists?

Yes, the inverse is unique.

What is the first step to find the inverse of a square matrix A?

Augment A with the identity matrix I.

What row operation goal must be achieved on the left side to find A⁻¹?

Transform the left side into the identity matrix I.

Where do you read the inverse matrix after row operations on the augmented matrix?

From the right side of the augmented matrix.

What are the three elementary row operations?

Interchange rows, multiply a row by a nonzero number, add a multiple of one row to another.

How does the inverse method solve AX = B?

It finds A⁻¹ and computes X = A⁻¹B.

When can the inverse method be applied to a system?

Only when the coefficient matrix is square and invertible.

What limitation does the inverse method have regarding solution types?

It cannot distinguish no-solution from infinitely-many-solution cases.

What is the first step in Gaussian elimination for AX = B?

Augment the coefficient matrix with the constant vector.

What is the goal of elementary row operations in Gaussian elimination?

To transform the coefficient matrix into identity form or reveal solution type.

What does a row of zeros with a nonzero constant signify?

The system has no solution.

What does an identity-form coefficient matrix indicate about solutions?

There is a unique solution.

What are the possible numbers of solutions for any linear system?

No solution, exactly one solution, or infinitely many solutions.

What is the first step to solve a word problem?

Represent an unknown with a letter.

How do you express other unknowns in word problems?

In terms of the first unknown letter.

What is the next step after expressing unknowns in word problems?

Translate the information into algebraic equations.

What equations represent the carpeting example's wool and nylon usage?

20X + 30Y = 1,500 and 40X + 30Y = 1,800.

What are the values of X and Y in the carpeting example?

X = 15 superior-grade rolls and Y = 40 quality-grade rolls.

What is a Markov chain?

A probabilistic model studying system evolution using transition probabilities.

Who introduced the Markov chain and when?

Andrew Markov, around 1907.

What can Markov chains predict about a system?

The probability of the system being in a particular state at a given time and its long-run equilibrium.

On what does a system's state in a given period depend?

Its state in the preceding period and the transition probabilities.

What are the assumptions of a Markov chain?

Constant transition probabilities, at most one change per period, and regular intervals.

What is a transition probability?

The likelihood a system in state i moves to state j next period.

How are future state vectors calculated in a Markov chain?

By multiplying the previous state vector by the transition matrix.

What is the initial distribution in the two-store example?

(0.8, 0.2).

What is the transition matrix in the two-store example?

$P=\begin{pmatrix}0.9&0.1\\0.2&0.8\end{pmatrix}$.

What is the long-run distribution in the two-store example?

Two-thirds customers at store 1 and one-third at store 2.

What is linear programming?

An optimization method allocating scarce resources with limiting conditions as inequalities or equations.

What objectives can a linear programming model maximize?

Profit, revenue, sales, or market share.

What objectives can a linear programming model minimize?

Cost, time, or distance.

What do decision variables represent in linear programming?

Unknown controllable quantities the decision maker must determine.

What are constraints in linear programming?

Restrictions limiting feasibility or attainability of a proposed action.

Give examples of constraints in linear programming.

Scarce resources, legal requirements, contracts, forecasts, orders, or policies.

What are parameters in linear programming?

Fixed numerical values specifying effects of decision variables on objectives and constraints.

What are the main assumptions of linear programming models?

Linearity, divisibility, certainty, and non-negativity.

What are the steps in formulating a linear programming model?

Define the problem, identify decision variables, specify the objective function, and identify constraints.

What is the objective in the microcomputer problem?

To maximize weekly profit by choosing quantities of two computer types to produce.

What is the objective function in the microcomputer linear programming model?

Maximize $Z=60X_1+50X_2$.

What are the constraints in the microcomputer linear programming model?

$4X_1+10X_2\le100$, $2X_1+X_2\le22$, $3X_1+3X_2\le39$, and $X_1,X_2\ge0$.

What does the switching-device model maximize?

$Z=8.25X_1+7.50X_2+7.80X_3$.

What constraints apply in the switching-device model?

Two assembly-time constraints of 450 minutes each, minimum production of 20 units per model, and non-negativity.

Why are non-negativity constraints required in linear programming?

Because negative quantities are unrealistic for decision variables.

Which method applies only to linear programming with two variables?

The graphical solution method.

Which method handles linear programming problems with more than two variables?

The algebraic simplex method.

How does the graphical method solve a linear programming problem?

By plotting constraints, finding the feasible region, and selecting the optimal point.

Where does at least one optimal solution occur in a linear programming problem?

At an extreme or corner point of the feasible region.

How do you identify the graphical optimum in linear programming?

Substitute corner point coordinates into the objective function and select the best value.

What is the optimal production and profit in the microcomputer model example?

Produce 9 units of type 1 and 4 units of type 2 for a profit of Br. 740.

What does slack represent in linear programming?

The unused amount of a scarce resource, equal to available minus used.

Which constraints have zero slack in the microcomputer solution?

Inspection time and storage space are binding constraints with zero slack.