TAMU MATH 140 Matrix Operations Flashcards

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Questions and Answers

What must matrices have for addition or subtraction?

Same number of columns

Different sizes

Same number of rows

Same size (correct)

What does matrix multiplication require?

Inner values must be equal, to give a matrix of the outer values

The commutative property states that for all mxn matrices, A+B = B+A.

True

What is the associative property of addition?

(a+b)+c=a+(b+c) Signup and view all the answers

What does the transpose of a matrix do?

MxN -> NxM Signup and view all the answers

Which of the following is true about properties of matrix transposition?

(A+B)^T = A^T + B Signup and view all the answers

What defines matrix equality?

Only equal if they are the same size and corresponding entries are equal Signup and view all the answers

What is one of the properties of matrix multiplication?

All of the above Signup and view all the answers

What is the equation of a horizontal line?

y = b Signup and view all the answers

What is the equation of a vertical line?

x = a Signup and view all the answers

What is the point slope form of a line?

y - y1 = m(x - x1) Signup and view all the answers

What is the slope-intercept form of a line?

y = mx + b Signup and view all the answers

What is the standard form of a linear equation?

Ax + By = C Signup and view all the answers

What does the linear depreciation formula represent?

V(t) = mt + b Signup and view all the answers

What is the total cost formula?

C(x) = (production cost per item)(quantity) + (fixed cost) = mx + F Signup and view all the answers

What does revenue R(x) equal?

R(x) = (price per item)(quantity) = px Signup and view all the answers

How is profit P(x) calculated?

P(x) = R(x) - C(x) Signup and view all the answers

What does demand D(x) represent?

D(x) = price/p(x) = mx + b, always has a negative slope Signup and view all the answers

What does supply S(x) represent?

S(x) = price/p(x) = mx + b, always has a positive slope Signup and view all the answers

What defines an independent system?

A system of equations that has exactly one solution. M1 not equal to M2. Signup and view all the answers

What is an inconsistent system?

A system of equations that has no solution. M1 = M2 and B1 is not equal to B2. Signup and view all the answers

What is a dependent system?

A system of equations that has infinitely many solutions. M1 = M2 and B1 = B2. Signup and view all the answers

What is the break-even point?

Where R(x) = C(x), ordered pair: (BE quantity, BE revenue) Signup and view all the answers

What is the equilibrium point?

Where S(x) = D(x), ordered pair: (EQ quantity, EQ price) Signup and view all the answers

What is the reduced row echelon form (RREF)?

Each leading 1 is to the right of all leading 1s in the rows above Signup and view all the answers

What is pivoting?

Process of obtaining a 1 in a location and making all other entries zeros in that column Signup and view all the answers

Study Notes

Matrix Operations

Matrix Addition/Subtraction: Only matrices of the same size can be added or subtracted.
Matrix Multiplication: Requires that inner dimensions are equal; resulting matrix dimensions are defined by the outer dimensions.
Commutative Property: For matrices A and B, A + B = B + A holds true.
Associative Property: The equation (a + b) + c = a + (b + c) applies to matrix addition.

Matrix Transposition

Transpose of a Matrix: Transforming a matrix of size MxN to NxM.
Properties of Matrix Transposition:
- If A is MxN, then A^T is NxM.
- Two successive transpositions return the original matrix: (A^T)^T = A.
- For any scalar k, (kA)^T = kA^T.
- The transpose of a sum: (A + B)^T = A^T + B^T.

Matrix Equality and Multiplication

Matrix Equality: Two matrices are equal if they have the same dimensions and matching corresponding entries.
Properties of Matrix Multiplication:
- Associative property: A(BC) = AB(C).
- Distributive property: A(B + C) = AB + AC and (A + B)C = AC + BC.

Linear Equations and Graphs

Horizontal Line: Represented by the equation y = b.
Vertical Line: Represented by the equation x = a.
Point-Slope Form: y - y1 = m(x - x1), a formula for the equation of a line considering a point and its slope.
Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept.
Standard Form: The linear equation format Ax + By = C.

Financial Applications

Linear Depreciation: V(t) = mt + b, where B is initial value, and M represents the depreciation rate.
Total Cost Function C(x): Represents costs as C(x) = (production cost per item)(quantity) + fixed cost, simplified to C(x) = mx + F.
Revenue Function R(x): The formula for revenue is R(x) = (price per item)(quantity), represented as R(x) = px.
Profit Function P(x): Profit is calculated as P(x) = R(x) - C(x).
Demand Function D(x): Typically linear with D(x) = mx + b, exhibiting a negative slope.
Supply Function S(x): Another linear function defined as S(x) = mx + b, with a positive slope.

Systems of Equations

Independent System: Contains exactly one solution; characterized by M1 ≠ M2.
Inconsistent System: No solution exists; indicated by M1 = M2 with B1 ≠ B2.
Dependent System: Infinitely many solutions, where M1 = M2 and B1 = B2.
Break Even Point: Occurs where total revenue R(x) equals total cost C(x), represented by the ordered pair (break even quantity, break even revenue).
Equilibrium Point: Defined by where supply S(x) equals demand D(x), represented as (equilibrium quantity, equilibrium price).

Reduced Row Echelon Form (RREF)

RREF Conditions:
- All zero rows positioned below non-zero rows.
- Each leading entry (first non-zero from the left) is 1 and is referred to as a leading 1.
- Each leading 1 must be right of leading 1s in higher rows.
- Each leading 1 is the only non-zero entry in its column.

Pivoting

Pivoting: A technique to obtain a leading 1 in a specific location while converting all other entries in that column to zero.

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Test your knowledge on matrix operations with these flashcards from TAMU MATH 140. Learn key concepts such as matrix addition, multiplication, and properties like commutative and associative. Perfect for exam preparation and understanding core mathematical principles related to matrices.