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Augmented Matrices for Linear Equations

• Augmented matrices simplify systems of linear equations.
• A vertical line separates the coefficients of variables from the constants.
• Each row in the matrix represents an equation.
• Coefficients of variables are on the left side, constants on the right.
• Missing variables have a coefficient of zero.

Row Operations for Matrix Reduction

• Interchange Rows: Swap any two rows.
• Multiply Row by a Nonzero Constant: Multiply a row by a non-zero scalar.
• Replace Row: Add a multiple of one row to another.
  • Example: Multiply a row by (say) 2, and add the result to another row to eliminate a variable.

Gaussian Elimination and Reduced Row-Echelon Form

• Gaussian Elimination: A method to reduce an augmented matrix to simplified form.
• Back-substitution: After Gaussian elimination, use the simplified form to solve for the variables systematically substituting back.
• Reduced Row-Echelon Form: The final simplified form of the matrix. It has 1s along the main diagonal and 0s below and above the 1s.

Steps in Gaussian Elimination

• Write the system of equations as an augmented matrix.
• Obtain a 1 in the first row, first column.
• Use row operations to create zeros below the 1 in the first column.
• Obtain a 1 in the second row, second column.
• Use row operations to create zeros below the 1 in the second column.
• Repeat until a 1 is in each of the rows, along the diagonal.
• Change back to equations.
• Use back-substitution to solve for variables.

Steps in Gauss-Jordan Elimination

• Follow the same steps as Gaussian elimination, but continue further till all entries above and below the 1s on the diagonal are zero.
• Use back-substitution to find the final solution set.