Gaussian Elimination

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What is Gaussian Elimination?

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A method for solving systems of linear equations by transforming the system's augmented matrix into Row Echelon Form (REF) or Reduced Row Echelon Form (RREF).

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What is Row Echelon Form (REF)?

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A matrix form where all non-zero rows are above any rows of all zeros, and the leading coefficient of a non-zero row is always to the right of the leading coefficient of the previous row.

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What is Reduced Row Echelon Form (RREF)?

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A matrix form where it is in Row Echelon Form, and each leading coefficient is 1 and is the only non-zero entry in its column.

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What does the term 'leading coefficient' refer to?

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The first non-zero number from the left in a row of a matrix.

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How do you represent a system of equations in matrix form?

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By creating an augmented matrix that includes the coefficients of the variables and the constants from the equations.

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What is the purpose of row operations in Gaussian Elimination?

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To manipulate the rows of the matrix to achieve Row Echelon Form or Reduced Row Echelon Form, making it easier to solve the system.

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What are the three types of row operations?

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1. Swapping two rows. 2. Multiplying a row by a non-zero scalar. 3. Adding or subtracting a multiple of one row to another row.

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What is the significance of a 3x4 matrix in a system of equations?

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It represents a system of 3 equations with 3 variables and an additional column for the constants.

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What does it mean if a system of equations has no solution?

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The lines represented by the equations are parallel and do not intersect.

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What does it mean if a system of equations has infinitely many solutions?

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The equations represent the same line or plane, leading to an infinite number of intersection points.