Augmented Matrices

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What is an augmented matrix?

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An augmented matrix is a matrix that represents a system of linear equations, including the coefficients of the variables and the constants from the equations.

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CCSS.HSA.REI.C.8

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How do you represent a system of equations as an augmented matrix?

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To represent a system of equations as an augmented matrix, write the coefficients of the variables in a matrix format and append the constants as an additional column.

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CCSS.HSA.REI.C.8

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What is the purpose of using augmented matrices in solving systems of equations?

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Augmented matrices simplify the process of solving systems of equations using methods like Gaussian elimination or row reduction.

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CCSS.HSA.REI.C.8

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What is Gaussian elimination?

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Gaussian elimination is a method for solving systems of linear equations by transforming the augmented matrix into row echelon form.

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CCSS.HSA.REI.C.6

CCSS.8.EE.C.8B

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What does it mean for a system of equations to have no solution?

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A system of equations has no solution if the equations represent parallel lines that never intersect.

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CCSS.8.EE.C.8A

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What does it mean for a system of equations to have infinitely many solutions?

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A system of equations has infinitely many solutions if the equations represent the same line or plane, meaning they overlap completely.

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CCSS.8.EE.C.8B

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What is row echelon form?

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

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What is reduced row echelon form?

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Reduced row echelon form is a form of a matrix 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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How do you determine if a system of equations is consistent?

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A system of equations is consistent if it has at least one solution, which can be determined by analyzing the augmented matrix.

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CCSS.HSA.REI.C.9

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What is the significance of the rank of a matrix?

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The rank of a matrix indicates the maximum number of linearly independent row or column vectors in the matrix, which helps determine the number of solutions in a system of equations.