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$[\begin{array}{c} 2 t \\ t \\ 3 t \end{array}]$

خطوة 1

Write as an augmented matrix for $A x = 0$ .

$[\begin{array}{c} 2 t & 0 \\ t & 0 \\ 3 t & 0 \end{array}]$

خطوة 2

أوجِد الصيغة الدرجية المختزلة صفيًا.

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خطوة 2.1

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

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خطوة 2.1.1

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

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

خطوة 2.1.2

بسّط $R_{1}$ .

$[\begin{array}{c} 1 & 0 \\ t & 0 \\ 3 t & 0 \end{array}]$

خطوة 2.2

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

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خطوة 2.2.1

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

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

خطوة 2.2.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & 0 \\ 0 & 0 \\ 3 t & 0 \end{array}]$

خطوة 2.3

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

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خطوة 2.3.1

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

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

خطوة 2.3.2

بسّط $R_{3}$ .

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

خطوة 3

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

$x = 0$

$0 = 0$

خطوة 4

Write a solution vector by solving in terms of the free variables in each row.

$[\begin{array}{c} x \end{array}] = [\begin{array}{c} 0 \end{array}]$

خطوة 5

Write as a solution set.

${[\begin{array}{c} 0 \end{array}]}$