Questions and Answers
What must matrices have for addition or subtraction?
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?
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What does the transpose of a matrix do?
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Which of the following is true about properties of matrix transposition?
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What defines matrix equality?
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What is one of the properties of matrix multiplication?
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What is the equation of a horizontal line?
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What is the equation of a vertical line?
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What is the point slope form of a line?
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What is the slope-intercept form of a line?
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What is the standard form of a linear equation?
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What does the linear depreciation formula represent?
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What is the total cost formula?
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What does revenue R(x) equal?
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How is profit P(x) calculated?
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What does demand D(x) represent?
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What does supply S(x) represent?
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What defines an independent system?
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What is an inconsistent system?
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What is a dependent system?
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What is the break-even point?
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What is the equilibrium point?
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What is the reduced row echelon form (RREF)?
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What is pivoting?
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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.
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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.
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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)
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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.