Reduced Row Echelon Form Of 4x5 Matrix - Justin K

Our Augmented Matrix is:

(1)
\begin{align} \left(\begin{array}{cc} 2 & -2 & 5 & 4 & 2 \\ 3 & 1 & -3 & 3 & 4 \\ 3 & -1 & 4 & -2 & -2 \\ -1 & 1 & 4 & 2 & 3 \end{array}\right) \end{align}

Find the Reduced Row Echelon Form

1. Select (1,1) as pivot point and divide that row by 2 to reduce the 2 in (1,1) to a 1.

(2)
\begin{align} \left(\begin{array}{cc} 1 & -1 & 5/2 & 2 & 1 \\ 3 & 1 & -3 & 3 & 4 \\ 3 & -1 & 4 & -2 & -2 \\ -1 & 1 & 4 & 2 & 3 \end{array}\right) \end{align}

2. Multiply second row by -1/3 and add the first row to it.

(3)
\begin{align} \left(\begin{array}{cc} 1 & -1 & 5/2 & 2 & 1 \\ 0 & -4/3 & 7/2 & 1 & -1/3 \\ 3 & -1 & 4 & -2 & -2 \\ -1 & 1 & 4 & 2 & 3 \end{array}\right) \end{align}

3. Multiply frist row by -3 and add it to row 3.

(4)
\begin{align} \left(\begin{array}{cc} 1 & -1 & 5/2 & 2 & 1 \\ 0 & -4/3 & 7/2 & 1 & -1/3 \\ 0 & 2 & 1/4 & -8 & -5 \\ -1 & 1 & 4 & 2 & 3 \end{array}\right) \end{align}

4. Divide row 2 by -4/3 to change (2,2) to a 1.

(5)
\begin{align} \left(\begin{array}{cc} 1 & -1 & 5/2 & 2 & 1 \\ 0 & -4/3 & 7/2 & 1 & -1/3 \\ 0 & 2 & 1/4 & -8 & -5 \\ -1 & 1 & 4 & 2 & 3 \end{array}\right) \end{align}

5. Add row 1 to row 3.

(6)
\begin{align} \left(\begin{array}{cc} 1 & -1 & 5/2 & 2 & 1 \\ 0 & 1 & -21/8 & -3/4 & 1/4 \\ 0 & 2 & 1/4 & -8 & -5 \\ 0 & 0 & 13/2 & 4 & 4 \end{array}\right) \end{align}

6. Multiply row 2 by -2 and add to row 3.

(7)
\begin{align} \left(\begin{array}{cc} 1 & -1 & 5/2 & 2 & 1 \\ 0 & 1 & -21/8 & -3/4 & 1/4 \\ 0 & 0 & 11/2 & -13/2 & -11/2 \\ 0 & 0 & 13/2 & 4 & 4 \end{array}\right) \end{align}

7. Divide row 3 by 11/2.

(8)
\begin{align} \left(\begin{array}{cc} 1 & -1 & 5/2 & 2 & 1 \\ 0 & 1 & -21/8 & -3/4 & 1/4 \\ 0 & 0 & 1 & -13/11 & -1 \\ 0 & 0 & 13/2 & 4 & 4 \end{array}\right) \end{align}

8. Multiply row 3 by -13/2 and add to row 4.

(9)
\begin{align} \left(\begin{array}{cc} 1 & -1 & 5/2 & 2 & 1 \\ 0 & 1 & -21/8 & -3/4 & 1/4 \\ 0 & 0 & 1 & -13/11 & -1 \\ 0 & 0 & 0 & 257/22 & 21/2 \end{array}\right) \end{align}

9. Divide row 4 by 257/22.

(10)
\begin{align} \left(\begin{array}{cc} 1 & -1 & 5/2 & 2 & 1 \\ 0 & 1 & -21/8 & -3/4 & 1/4 \\ 0 & 0 & 1 & -13/11 & -1 \\ 0 & 0 & 0 & 1 & 231/257 \end{array}\right) \end{align}

10. Add row 2 to row 1.

(11)
\begin{align} \left(\begin{array}{cc} 1 & 0 & -1/8 & 5/4 & 5/4 \\ 0 & 1 & -21/8 & -3/4 & 1/4 \\ 0 & 0 & 1 & -13/11 & -1 \\ 0 & 0 & 0 & 1 & 231/257 \end{array}\right) \end{align}

11. Multiply row 3 by 1/8 and add to row 1.

(12)
\begin{align} \left(\begin{array}{cc} 1 & 0 & 0 & 97/88 & 9/8 \\ 0 & 1 & -21/8 & -3/4 & 1/4 \\ 0 & 0 & 1 & -13/11 & -1 \\ 0 & 0 & 0 & 1 & 231/257 \end{array}\right) \end{align}

12. Multiply row 3 by 21/8 and add to row 2.

(13)
\begin{align} \left(\begin{array}{cc} 1 & 0 & 0 & 97/88 & 9/8 \\ 0 & 1 & 0 & -339/88 & -19/8 \\ 0 & 0 & 1 & -13/11 & -1 \\ 0 & 0 & 0 & 1 & 231/257 \end{array}\right) \end{align}

13. Multiply row 4 by -97/88 and add to row 1.

(14)
\begin{align} \left(\begin{array}{cc} 1 & 0 & 0 & 0 & 69/514 \\ 0 & 1 & 0 & -339/88 & -19/8 \\ 0 & 0 & 1 & -13/11 & -1 \\ 0 & 0 & 0 & 1 & 231/257 \end{array}\right) \end{align}

14. Multiply row 4 by 339/88 an add to row 2.

(15)
\begin{align} \left(\begin{array}{cc} 1 & 0 & 0 & 0 & 69/514 \\ 0 & 1 & 0 & 0 & 559/514 \\ 0 & 0 & 1 & -13/11 & -1 \\ 0 & 0 & 0 & 1 & 231/257 \end{array}\right) \end{align}

15. Multiply row 4 by 13/11 and add to row 3.

(16)
\begin{align} \left(\begin{array}{cc} 1 & 0 & 0 & 0 & 69/514 \\ 0 & 1 & 0 & 0 & 559/514 \\ 0 & 0 & 1 & 0 & 16/257 \\ 0 & 0 & 0 & 1 & 231/257 \end{array}\right) \end{align}

Wow! What a lot of work! Using Octave, I arrived at these same numbers with a deviation of approximately .02, well within an acceptable margin. Octave obviously rounded the numbers given. Below are the numbers found using Octave. As can be seen, the answers differ by mere tenths or hundredths.

(17)
\begin{align} \left(\begin{array}{cc} 1 & 0 & 0 & 0 & .17513 \\ 0 & 1 & 0 & 0 & 1.11421 \\ 0 & 0 & 1 & 0 & .08122 \\ 0 & 0 & 0 & 1 & .86802 \end{array}\right) \end{align}
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