Đại số tuyến tính Ví dụ

$[\begin{array}{c} - 2 & - 1 & 8 & 3 \\ 0 & - 6 & 4 & - 2 \\ 8 & 6 & 3 & 25 \end{array}]$

Bước 1

Tìm dạng ma trận hàng bậc thang rút gọn.

Nhấp để xem thêm các bước...

Bước 1.1

Multiply each element of $R_{1}$ by $- \frac{1}{2}$ to make the entry at $1, 1$ a $1$ .

Nhấp để xem thêm các bước...

Bước 1.1.1

Multiply each element of $R_{1}$ by $- \frac{1}{2}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} - \frac{1}{2} \cdot - 2 & - \frac{1}{2} \cdot - 1 & - \frac{1}{2} \cdot 8 & - \frac{1}{2} \cdot 3 \\ 0 & - 6 & 4 & - 2 \\ 8 & 6 & 3 & 25 \end{array}]$

Bước 1.1.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - 4 & - \frac{3}{2} \\ 0 & - 6 & 4 & - 2 \\ 8 & 6 & 3 & 25 \end{array}]$

Bước 1.2

Perform the row operation $R_{3} = R_{3} - 8 R_{1}$ to make the entry at $3, 1$ a $0$ .

Nhấp để xem thêm các bước...

Bước 1.2.1

Perform the row operation $R_{3} = R_{3} - 8 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - 4 & - \frac{3}{2} \\ 0 & - 6 & 4 & - 2 \\ 8 - 8 \cdot 1 & 6 - 8 (\frac{1}{2}) & 3 - 8 \cdot - 4 & 25 - 8 (- \frac{3}{2}) \end{array}]$

Bước 1.2.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - 4 & - \frac{3}{2} \\ 0 & - 6 & 4 & - 2 \\ 0 & 2 & 35 & 37 \end{array}]$

Bước 1.3

Multiply each element of $R_{2}$ by $- \frac{1}{6}$ to make the entry at $2, 2$ a $1$ .

Nhấp để xem thêm các bước...

Bước 1.3.1

Multiply each element of $R_{2}$ by $- \frac{1}{6}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - 4 & - \frac{3}{2} \\ - \frac{1}{6} \cdot 0 & - \frac{1}{6} \cdot - 6 & - \frac{1}{6} \cdot 4 & - \frac{1}{6} \cdot - 2 \\ 0 & 2 & 35 & 37 \end{array}]$

Bước 1.3.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - 4 & - \frac{3}{2} \\ 0 & 1 & - \frac{2}{3} & \frac{1}{3} \\ 0 & 2 & 35 & 37 \end{array}]$

Bước 1.4

Perform the row operation $R_{3} = R_{3} - 2 R_{2}$ to make the entry at $3, 2$ a $0$ .

Nhấp để xem thêm các bước...

Bước 1.4.1

Perform the row operation $R_{3} = R_{3} - 2 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - 4 & - \frac{3}{2} \\ 0 & 1 & - \frac{2}{3} & \frac{1}{3} \\ 0 - 2 \cdot 0 & 2 - 2 \cdot 1 & 35 - 2 (- \frac{2}{3}) & 37 - 2 (\frac{1}{3}) \end{array}]$

Bước 1.4.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - 4 & - \frac{3}{2} \\ 0 & 1 & - \frac{2}{3} & \frac{1}{3} \\ 0 & 0 & \frac{109}{3} & \frac{109}{3} \end{array}]$

Bước 1.5

Multiply each element of $R_{3}$ by $\frac{3}{109}$ to make the entry at $3, 3$ a $1$ .

Nhấp để xem thêm các bước...

Bước 1.5.1

Multiply each element of $R_{3}$ by $\frac{3}{109}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - 4 & - \frac{3}{2} \\ 0 & 1 & - \frac{2}{3} & \frac{1}{3} \\ \frac{3}{109} \cdot 0 & \frac{3}{109} \cdot 0 & \frac{3}{109} \cdot \frac{109}{3} & \frac{3}{109} \cdot \frac{109}{3} \end{array}]$

Bước 1.5.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - 4 & - \frac{3}{2} \\ 0 & 1 & - \frac{2}{3} & \frac{1}{3} \\ 0 & 0 & 1 & 1 \end{array}]$

Bước 1.6

Perform the row operation $R_{2} = R_{2} + \frac{2}{3} R_{3}$ to make the entry at $2, 3$ a $0$ .

Nhấp để xem thêm các bước...

Bước 1.6.1

Perform the row operation $R_{2} = R_{2} + \frac{2}{3} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - 4 & - \frac{3}{2} \\ 0 + \frac{2}{3} \cdot 0 & 1 + \frac{2}{3} \cdot 0 & - \frac{2}{3} + \frac{2}{3} \cdot 1 & \frac{1}{3} + \frac{2}{3} \cdot 1 \\ 0 & 0 & 1 & 1 \end{array}]$

Bước 1.6.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - 4 & - \frac{3}{2} \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \end{array}]$

Bước 1.7

Perform the row operation $R_{1} = R_{1} + 4 R_{3}$ to make the entry at $1, 3$ a $0$ .

Nhấp để xem thêm các bước...

Bước 1.7.1

Perform the row operation $R_{1} = R_{1} + 4 R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 + 4 \cdot 0 & \frac{1}{2} + 4 \cdot 0 & - 4 + 4 \cdot 1 & - \frac{3}{2} + 4 \cdot 1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \end{array}]$

Bước 1.7.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & \frac{5}{2} \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \end{array}]$

Bước 1.8

Perform the row operation $R_{1} = R_{1} - \frac{1}{2} R_{2}$ to make the entry at $1, 2$ a $0$ .

Nhấp để xem thêm các bước...

Bước 1.8.1

Perform the row operation $R_{1} = R_{1} - \frac{1}{2} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{1}{2} \cdot 0 & \frac{1}{2} - \frac{1}{2} \cdot 1 & 0 - \frac{1}{2} \cdot 0 & \frac{5}{2} - \frac{1}{2} \cdot 1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \end{array}]$

Bước 1.8.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \end{array}]$

Bước 2

The pivot positions are the locations with the leading $1$ in each row. The pivot columns are the columns that have a pivot position.

Pivot Positions: $a_{11}, a_{22},$ and $a_{33}$

Pivot Columns: $1, 2,$ and $3$