Đại số tuyến tính Ví dụ

$[\begin{array}{c} 1 & 2 & 3 & 4 \\ 2 & 1 & 4 & 5 \\ 1 & 5 & 5 & 7 \\ 8 & 1 & 14 & 17 \end{array}]$

Bước 1

Tìm dạng ma trận hàng bậc thang rút gọn.

Nhấp để xem thêm các bước...

Bước 1.1

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

Nhấp để xem thêm các bước...

Bước 1.1.1

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

$[\begin{array}{c} 1 & 2 & 3 & 4 \\ 2 - 2 \cdot 1 & 1 - 2 \cdot 2 & 4 - 2 \cdot 3 & 5 - 2 \cdot 4 \\ 1 & 5 & 5 & 7 \\ 8 & 1 & 14 & 17 \end{array}]$

Bước 1.1.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 \\ 0 & - 3 & - 2 & - 3 \\ 1 & 5 & 5 & 7 \\ 8 & 1 & 14 & 17 \end{array}]$

Bước 1.2

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

Nhấp để xem thêm các bước...

Bước 1.2.1

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

$[\begin{array}{c} 1 & 2 & 3 & 4 \\ 0 & - 3 & - 2 & - 3 \\ 1 - 1 & 5 - 2 & 5 - 3 & 7 - 4 \\ 8 & 1 & 14 & 17 \end{array}]$

Bước 1.2.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 \\ 0 & - 3 & - 2 & - 3 \\ 0 & 3 & 2 & 3 \\ 8 & 1 & 14 & 17 \end{array}]$

Bước 1.3

Perform the row operation $R_{4} = R_{4} - 8 R_{1}$ to make the entry at $4, 1$ a $0$ .

Nhấp để xem thêm các bước...

Bước 1.3.1

Perform the row operation $R_{4} = R_{4} - 8 R_{1}$ to make the entry at $4, 1$ a $0$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 \\ 0 & - 3 & - 2 & - 3 \\ 0 & 3 & 2 & 3 \\ 8 - 8 \cdot 1 & 1 - 8 \cdot 2 & 14 - 8 \cdot 3 & 17 - 8 \cdot 4 \end{array}]$

Bước 1.3.2

Rút gọn $R_{4}$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 \\ 0 & - 3 & - 2 & - 3 \\ 0 & 3 & 2 & 3 \\ 0 & - 15 & - 10 & - 15 \end{array}]$

Bước 1.4

Multiply each element of $R_{2}$ by $- \frac{1}{3}$ to make the entry at $2, 2$ a $1$ .

Nhấp để xem thêm các bước...

Bước 1.4.1

Multiply each element of $R_{2}$ by $- \frac{1}{3}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 \\ - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot - 3 & - \frac{1}{3} \cdot - 2 & - \frac{1}{3} \cdot - 3 \\ 0 & 3 & 2 & 3 \\ 0 & - 15 & - 10 & - 15 \end{array}]$

Bước 1.4.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 \\ 0 & 1 & \frac{2}{3} & 1 \\ 0 & 3 & 2 & 3 \\ 0 & - 15 & - 10 & - 15 \end{array}]$

Bước 1.5

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

Nhấp để xem thêm các bước...

Bước 1.5.1

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

$[\begin{array}{c} 1 & 2 & 3 & 4 \\ 0 & 1 & \frac{2}{3} & 1 \\ 0 - 3 \cdot 0 & 3 - 3 \cdot 1 & 2 - 3 (\frac{2}{3}) & 3 - 3 \cdot 1 \\ 0 & - 15 & - 10 & - 15 \end{array}]$

Bước 1.5.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 \\ 0 & 1 & \frac{2}{3} & 1 \\ 0 & 0 & 0 & 0 \\ 0 & - 15 & - 10 & - 15 \end{array}]$

Bước 1.6

Perform the row operation $R_{4} = R_{4} + 15 R_{2}$ to make the entry at $4, 2$ a $0$ .

Nhấp để xem thêm các bước...

Bước 1.6.1

Perform the row operation $R_{4} = R_{4} + 15 R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 \\ 0 & 1 & \frac{2}{3} & 1 \\ 0 & 0 & 0 & 0 \\ 0 + 15 \cdot 0 & - 15 + 15 \cdot 1 & - 10 + 15 (\frac{2}{3}) & - 15 + 15 \cdot 1 \end{array}]$

Bước 1.6.2

Rút gọn $R_{4}$ .

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

Bước 1.7

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

Nhấp để xem thêm các bước...

Bước 1.7.1

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

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

Bước 1.7.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & 0 & \frac{5}{3} & 2 \\ 0 & 1 & \frac{2}{3} & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

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: $a_{11}$ and $a_{22}$

Pivot Columns: $1$ and $2$

Bước 3

The rank is the number of pivot columns.

$2$