Euclidean m -Space & Linear Equations. Row Reduction of Linear Systems. Matrix. A matrix is a rectangular array of numbers. This array is enclosed in brackets or parentheses. Each number in a matrix is called and element or entry of that matrix.
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Row Reduction of Linear Systems
A matrix is a rectangular array of numbers.
This array is enclosed in brackets or parentheses.
Each number in a matrix is called and element or entry of that matrix.
Each horizontal line of numbers in a matrix is called a row.
Each vertical line of numbers in a matrix is called a column.
Matrix arise from a linear system in standard form by deleting all unknowns operations and equal signs.
A vertical line is drawn along the equal signs to separate the constant column.
When an unknown is missing in an equation, the coefficient zero is inserted in the augmented matrix.
Multiply a row by a nonzero scalar.
Interchange the positions of two rows.
Replace a row by the sum of itself and a multiple of another row.
x1 + x2 + 2x3 = 8
-x1 - 2x2 + 3x3 = 1
3x1 - 7x2 + 4x3 = 10
When one matrix can be obtained from another matrix by a finite sequence of row operations, the two matrices are said to be row-equivalent.
The underlying linear systems of two row-equivalent matrices have same solutions.
A matrix is in row reduced echelon form if it satisfies the following conditions:
The process of transforming an augmented matrix to an equivalent row-reduced echelon form is called the Gauss-Jordan Elimination.
A matrix is in row echelon form if it satisfies the following conditions: