Đại số tuyến tính Ví dụ

$2 a + b - g = 4$ , $b + g + h = 4$ , $a - d + 2 g = 0$

Bước 1

Write the system as a matrix.

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

Bước 2

Tìm dạng ma trận hàng bậc thang rút gọn.

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Bước 2.1

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

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Bước 2.1.1

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

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

Bước 2.1.2

Rút gọn $R_{1}$ .

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

Bước 2.2

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

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Bước 2.2.1

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

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

Bước 2.2.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & - \frac{1}{2} & 0 & 2 \\ 0 & 1 & 0 & 1 & 1 & 4 \\ 0 & - \frac{1}{2} & - 1 & \frac{5}{2} & 0 & - 2 \end{array}]$

Bước 2.3

Perform the row operation $R_{3} = R_{3} + \frac{1}{2} R_{2}$ to make the entry at $3, 2$ a $0$ .

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Bước 2.3.1

Perform the row operation $R_{3} = R_{3} + \frac{1}{2} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & - \frac{1}{2} & 0 & 2 \\ 0 & 1 & 0 & 1 & 1 & 4 \\ 0 + \frac{1}{2} \cdot 0 & - \frac{1}{2} + \frac{1}{2} \cdot 1 & - 1 + \frac{1}{2} \cdot 0 & \frac{5}{2} + \frac{1}{2} \cdot 1 & 0 + \frac{1}{2} \cdot 1 & - 2 + \frac{1}{2} \cdot 4 \end{array}]$

Bước 2.3.2

Rút gọn $R_{3}$ .

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

Bước 2.4

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

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Bước 2.4.1

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

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

Bước 2.4.2

Rút gọn $R_{3}$ .

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

Bước 2.5

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

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Bước 2.5.1

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

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

Bước 2.5.2

Rút gọn $R_{1}$ .

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

Bước 3

Use the result matrix to declare the final solution to the system of equations.

$a - g - \frac{1}{2} h = 0$

$b + g + h = 4$

$d - 3 g - \frac{1}{2} h = 0$

Bước 4

The solution is the set of ordered pairs that make the system true.

$(g + \frac{h}{2}, 4 - g - h, 3 g + \frac{h}{2}, g, h)$