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1.2 Row Reduction and Echelon Forms

\[\begin{bmatrix} 1 & 3 & 5 & 7 \\ 3 & 5 & 7 & 9 \\ 5 & 7 & 9 & 1 \end{bmatrix}\]

\(-3R_1 + R_2\)
\(-5R_1 + R_3\)

\[\begin{bmatrix} 1 & 3 & 5 & 7 \\ 0 & -4 & -8 & -12 \\ 0 & -8 & -16 & -34 \end{bmatrix}\]

\(-2R_2 + R_3\)

\[\begin{bmatrix} 1 & 3 & 5 & 7 \\ 0 & -4 & -8 & -12 \\ 0 & 0 & 0 & -10 \end{bmatrix}\]

this is now in Row Echelon Form

ladders / stairs

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this matrix is now in row echelon form because

  1. any zero row is at bottom
  2. the leading entry of each row is in a column to the right of the leading entry of the row above it.
  3. all entries in a column below a leading entry are zeros.
\[\begin{bmatrix} 1 & 3 & 5 & 7 \\ 0 & -4 & -8 & -12 \\ 0 & 0 & 0 & -10 \end{bmatrix}\]

let's go further and make more entries zeros

\[\begin{bmatrix} 1 & 3 & 5 & 7 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 1 \end{bmatrix}\]

\(-3R_2 + R_1\)

\[\begin{bmatrix} 1 & 0 & -1 & -2 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 1 \end{bmatrix}\]

\(-3R_3 + R_2\)
\(2R_3 + R_1\)

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\[ \begin{bmatrix} 1 & 0 & -1 & 0 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \]

this is now in Reduced Row Echelon Form

this matrix is in reduced row echelon form because (must satisfy conditions 1-3 above)

  1. leading entry of each nonzero row is 1
  2. each leading entry is the only nonzero element in its column.

the reduced row echelon form of a matrix is unique

\[ \begin{bmatrix} 1 & 0 & 5 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 \end{bmatrix} \]

neither because zero row not at bottom

\[ \begin{bmatrix} 1 & 0 & 3 & 2 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & 5 \end{bmatrix} \]

row echelon

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\[ \begin{bmatrix} 0 & 1 & -2 & 0 & 1 \\ 0 & 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix} \]

reduced row echelon

back to

\[ \begin{bmatrix} 1 & 0 & -1 & 0 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \]

in reduced row echelon form

the locations of the leading 1's are called pivot positions

the columns containing leading 1's are pivot columns

original matrix:

\[ \begin{bmatrix} 1 & 3 & 5 & 7 \\ 3 & 5 & 7 & 9 \\ 5 & 7 & 9 & 1 \end{bmatrix} \]

the elements correspond to the pivot positions in reduced echelon form are the ones used to make other rows in their column zeros

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Row Reduction and Basic Variables

The row reduction algorithm can be done to ANY matrix, not just augmented matrix of system.

In a system, the number of pivots (not in last column) corresponds to the number of basic variables in a system.

Example

\[\begin{aligned} 3x_1 - 4x_2 + 2x_3 &= 0 \\ -9x_1 + 12x_2 - 6x_3 &= 0 \\ -6x_1 + 8x_2 - 4x_3 &= 0 \end{aligned}\]
\[\begin{bmatrix} 3 & -4 & 2 & 0 \\ -9 & 12 & -6 & 0 \\ -6 & 8 & -4 & 0 \end{bmatrix}\]

row equivalent to

\[\sim \begin{bmatrix} 3 & -4 & 2 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \sim \begin{bmatrix} 1 & -4/3 & 2/3 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}\]

sys has 3 variables, 1 pivot (corresponds to \(x_1\))

\(\Rightarrow x_2\) and \(x_3\) are free variables

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\(x_1\) (pivot column) is a basic variable

Solution:

\[\begin{aligned} x_1 - \frac{4}{3}x_2 + \frac{2}{3}x_3 &= 0 \\ x_2 & \text{ free (arbitrary)} \\ x_3 & \text{ free ( " )} \end{aligned}\]

Parametric description:

\[\begin{aligned} x_1 &= \frac{4}{3}x_2 - \frac{2}{3}x_3 \\ x_2 & \text{ free} \\ x_3 & \text{ free} \end{aligned}\]

# of zero rows = # of free variables

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Linear Systems Examples

Example 1

The augmented matrix of some system reduces to:

\[ \begin{bmatrix} \boxed{1} & 0 & 0 & | & 1 \\ 0 & \boxed{1} & 0 & | & -1 \\ 0 & 0 & \boxed{1} & | & 1 \end{bmatrix} \]

3 variables, 3 pivots. One solution: \( (1, -1, 1) \)

Example 2

The augmented matrix of some system reduces to:

\[ \begin{bmatrix} \boxed{1} & 0 & 0 & 0 \\ 0 & \boxed{1} & 0 & 0 \\ 0 & 0 & 0 & \boxed{1} \\ 0 & 0 & 0 & 0 \end{bmatrix} \]

"overdetermined system"

more eqs. than variables

3 variables, 3 pivots BUT one in right most column.

Row 3 \( \rightarrow 0 = 1 \)

no solution (inconsistent system)

Even though there is a zero row, system is inconsistent, no solution.

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Any time there is a pivot in the right most column of the reduced echelon form of an augmented matrix of a system \( \Rightarrow \) inconsistent system, no solution.