Question

in exercises 33-36, write the matrix in row-echelon form. remember that the row-echelon form of a matrix is not unique.

1. $\begin{bmatrix} 1&2&3&0\\\\ -1&4&0&-5\\\\ 2&6&3&10 end{bmatrix}$
2. $\begin{bmatrix} 1&2&-1&3\\\\ 3&7&-5&14\\\\ -2&-1&-3&8 end{bmatrix}$
3. $\begin{bmatrix} 1&-1&-1&1\\\\ 5&-4&1&8\\\\ -6&8&18&0 end{bmatrix}$

Explanation:

Step1: Eliminate $R_2$ first column

$R_2 = R_2 + R_1$

$$\begin{bmatrix} 1 & 2 & 3 & 0 \\ 0 & 6 & 3 & -5 \\ 2 & 6 & 3 & 10 \end{bmatrix}$$

Step2: Eliminate $R_3$ first column

$R_3 = R_3 - 2R_1$

$$\begin{bmatrix} 1 & 2 & 3 & 0 \\ 0 & 6 & 3 & -5 \\ 0 & 2 & -3 & 10 \end{bmatrix}$$

Step3: Scale $R_2$ to simplify

$R_2 = \frac{1}{6}R_2$

$$\begin{bmatrix} 1 & 2 & 3 & 0 \\ 0 & 1 & \frac{1}{2} & -\frac{5}{6} \\ 0 & 2 & -3 & 10 \end{bmatrix}$$

Step4: Eliminate $R_3$ second column

$R_3 = R_3 - 2R_2$

$$\begin{bmatrix} 1 & 2 & 3 & 0 \\ 0 & 1 & \frac{1}{2} & -\frac{5}{6} \\ 0 & 0 & -4 & \frac{35}{3} \end{bmatrix}$$

Step5: Scale $R_3$ for leading 1

$R_3 = -\frac{1}{4}R_3$

$$\begin{bmatrix} 1 & 2 & 3 & 0 \\ 0 & 1 & \frac{1}{2} & -\frac{5}{6} \\ 0 & 0 & 1 & -\frac{35}{12} \end{bmatrix}$$

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Step1: Eliminate $R_2$ first column

$R_2 = R_2 - 3R_1$

$$\begin{bmatrix} 1 & 2 & -1 & 3 \\ 0 & 1 & -2 & 5 \\ -2 & -1 & -3 & 8 \end{bmatrix}$$

Step2: Eliminate $R_3$ first column

$R_3 = R_3 + 2R_1$

$$\begin{bmatrix} 1 & 2 & -1 & 3 \\ 0 & 1 & -2 & 5 \\ 0 & 3 & -5 & 14 \end{bmatrix}$$

Step3: Eliminate $R_3$ second column

$R_3 = R_3 - 3R_2$

$$\begin{bmatrix} 1 & 2 & -1 & 3 \\ 0 & 1 & -2 & 5 \\ 0 & 0 & 1 & -1 \end{bmatrix}$$

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Step1: Eliminate $R_2$ first column

$R_2 = R_2 - 5R_1$

$$\begin{bmatrix} 1 & -1 & -1 & 1 \\ 0 & 1 & 6 & 3 \\ -6 & 8 & 18 & 0 \end{bmatrix}$$

Step2: Eliminate $R_3$ first column

$R_3 = R_3 + 6R_1$

$$\begin{bmatrix} 1 & -1 & -1 & 1 \\ 0 & 1 & 6 & 3 \\ 0 & 2 & 12 & 6 \end{bmatrix}$$

Step3: Eliminate $R_3$ second column

$R_3 = R_3 - 2R_2$

$$\begin{bmatrix} 1 & -1 & -1 & 1 \\ 0 & 1 & 6 & 3 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$

Answer:

33.

$$\begin{bmatrix} 1 & 2 & 3 & 0 \\ 0 & 1 & \frac{1}{2} & -\frac{5}{6} \\ 0 & 0 & 1 & -\frac{35}{12} \end{bmatrix}$$

34.

$$\begin{bmatrix} 1 & 2 & -1 & 3 \\ 0 & 1 & -2 & 5 \\ 0 & 0 & 1 & -1 \end{bmatrix}$$

35.

$$\begin{bmatrix} 1 & -1 & -1 & 1 \\ 0 & 1 & 6 & 3 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$