| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To find the determinant of the matrix \[ \left( \begin{array}{cc} 0 & -4 \\ 0 & 3 \\ \end{array} \right) \] we use the formula for the determinant of a 2x2 matrix \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\), which is given by: \[ \text{det} = ad - bc \] For the given matrix, we have: - \(a = 0\) - \(b = -4\) - \(c = 0\) - \(d = 3\) Substituting these values into the determinant formula, we get: \[ \text{det} = (0 \cdot 3) - (-4 \cdot 0) = 0 - 0 = 0 \] Thus, the determinant of the matrix is: \[ \boxed{0} \] |
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| deepseek-r1 | To find the determinant of the matrix \[ \left( \begin{array}{cc} 0 & -4 \\ 0 & 3 \\ \end{array} \right), \] we use the formula for the determinant of a 2x2 matrix \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\), which is \(ad - bc\). For the given matrix, the entries are \(a = 0\), \(b = -4\), \(c = 0\), and \(d = 3\). Calculating the determinant step by step: 1. Compute \(ad\): \(0 \times 3 = 0\). 2. Compute \(bc\): \(-4 \times 0 = 0\). 3. Subtract the results: \(0 - 0 = 0\). Additionally, since the first column of the matrix consists entirely of zeros, the determinant is zero by the property that a matrix with an entire column (or row) of zeros has a determinant of zero. Thus, the determinant of the matrix is \(\boxed{0}\). |
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| gpto1 | To calculate the determinant of the given \(2 \times 2\) matrix: \[ \begin{pmatrix} 0 & -4 \\ 0 & 3 \end{pmatrix} \] We use the formula for the determinant of a \(2 \times 2\) matrix: \[ \text{Determinant} = (a)(d) - (b)(c) \] Where \(a\), \(b\), \(c\), and \(d\) are the elements of the matrix: \[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \] For our matrix: \[ a = 0,\quad b = -4,\quad c = 0,\quad d = 3 \] Plugging these values into the determinant formula: \[ \text{Determinant} = (0)(3) - (-4)(0) = 0 - 0 = 0 \] So, the determinant of the matrix is 0. \[\boxed{0}\] |
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| gpt-4o-2024-08-06 | To find the determinant of a \(2 \times 2\) matrix, we use the formula for a matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \), which is given by: \[ \text{det} = ad - bc \] For the matrix \[ \begin{pmatrix} 0 & -4 \\ 0 & 3 \end{pmatrix} \] we have \( a = 0 \), \( b = -4 \), \( c = 0 \), and \( d = 3 \). Substituting these values into the determinant formula, we get: \[ \text{det} = (0)(3) - (0)(-4) = 0 - 0 = 0 \] Thus, the determinant of the matrix is \(\boxed{0}\). |
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| DeepSeek-Coder-V2-Lite-Instruct | The determinant of a 2x2 matrix \(\left(\begin{array}{cc} a & b \\ c & d \end{array}\right)\) is calculated as \(ad - bc\). For the given matrix \(\left(\begin{array}{cc} 0 & -4 \\ 0 & 3 \end{array}\right)\): - \(a = 0\) - \(b = -4\) - \(c = 0\) - \(d = 3\) The determinant is calculated as: \[ \text{det} = (0 \cdot 3) - (0 \cdot -4) = 0 - 0 = 0 \] Thus, the determinant of the matrix is \(\boxed{0}\). |
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| Meta-Llama-3.1-405B-Instruct-Turbo | ## Step 1: Recall the formula for the determinant of a 2x2 matrix The determinant of a 2x2 matrix $\left(\begin{array}{cc} a & b \\ c & d \\ \end{array}\right)$ is given by the formula $ad - bc$. ## Step 2: Apply the formula to the given matrix For the given matrix $\left(\begin{array}{cc} 0 & -4 \\ 0 & 3 \\ \end{array}\right)$, we have $a = 0$, $b = -4$, $c = 0$, and $d = 3$. Plugging these values into the formula, we get $(0)(3) - (-4)(0)$. ## Step 3: Simplify the expression $(0)(3) - (-4)(0) = 0 - 0 = 0$. The final answer is: $\boxed{0}$ |
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| Meta-Llama-3.1-70B-Instruct-Turbo | ## Step 1: Recall the formula for the determinant of a 2x2 matrix The determinant of a 2x2 matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is given by the formula $ad - bc$. ## Step 2: Apply the formula to the given matrix Using the formula, we can calculate the determinant of the given matrix $\begin{pmatrix} 0 & -4 \\ 0 & 3 \end{pmatrix}$ as $(0)(3) - (-4)(0)$. ## Step 3: Simplify the expression The expression $(0)(3) - (-4)(0)$ simplifies to $0 - 0$, which equals $0$. The final answer is: $\boxed{0}$ |
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| Qwen2-72B-Instruct | The determinant of a 2x2 matrix $\left(\begin{array}{cc}a & b \\c & d \\\end{array}\right)$ is given by $ad - bc$. For the given matrix $\left(\begin{array}{cc}0 & -4 \\0 & 3 \\\end{array}\right)$, we have $a = 0$, $b = -4$, $c = 0$, and $d = 3$. Thus, the determinant is $0*3 - (-4)*0 = 0$. Therefore, the answer is $\boxed{0}$. |
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| baichuan4 | To find the determinant of a matrix, we use the following formula: For a 2x2 matrix $\left(\begin{array}{cc}a & b \\c & d \\\end{array}\right)$, the determinant is $ad - bc$. Given the matrix $\left(\begin{array}{cc}0 & -4 \\0 & 3 \\\end{array}\right)$, we can apply this formula with $a=0$, $b=-4$, $c=0$, and $d=3$. The determinant is then calculated as: $ad - bc = (0)(3) - (0)(-4) = 0 - 0 = 0$. Therefore, the determinant of the given matrix is $0$. Putting the final answer in a box, we have: $\boxed{0}$. |
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| doubao | The determinant of a 2x2 matrix $\begin{pmatrix}a & b\\c & d\end{pmatrix}$ is $ad-bc$. For the given matrix $\begin{pmatrix}0 & -4\\0 & 3\end{pmatrix}$, the determinant is $0\times3-0\times(-4)=\boxed{0}$. | 0 |
| gpto1 | To calculate the determinant of the given \(2 \times 2\) matrix: \[ \begin{pmatrix} 0 & -4 \\ 0 & 3 \end{pmatrix} \] We use the formula for the determinant of a \(2 \times 2\) matrix: \[ \text{Determinant} = (a)(d) - (b)(c) \] Where \(a\), \(b\), \(c\), and \(d\) are the elements of the matrix: \[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \] For our matrix: \[ a = 0,\quad b = -4,\quad c = 0,\quad d = 3 \] Plugging these values into the determinant formula: \[ \text{Determinant} = (0)(3) - (-4)(0) = 0 - 0 = 0 \] So, the determinant of the matrix is 0. \[\boxed{0}\] |
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| kimi | To find the determinant of the given 2x2 matrix, we can use the formula for the determinant of a 2x2 matrix: $\text{det}(A) = ad - bc$ where A is the given matrix: $A = \left(\begin{array}{cc} 0 & -4 \\ 0 & 3 \\ \end{array}\right)$ In this case, a = 0, b = -4, c = 0, and d = 3. Plugging these values into the formula, we get: $\text{det}(A) = (0)(3) - (-4)(0) = 0$ So, the determinant of the given matrix is: $\boxed{0}$ |
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| minimax | To find the determinant of a 2x2 matrix, you can use the formula: $$ \text{det}\left(\begin{array}{cc} a & b \\ c & d \\ \end{array}\right) = ad - bc $$ For the given matrix: $$ \left(\begin{array}{cc} 0 & -4 \\ 0 & 3 \\ \end{array}\right) $$ The determinant is: $$ \text{det} = (0)(3) - (-4)(0) = 0 - 0 = 0 $$ So the determinant of the matrix is: $$ \boxed{0} $$ |
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| wenxinyiyan | 这是一个关于矩阵的问题。我们要找出一个2x2矩阵的行列式。 给定的矩阵是： | 0 -4 | | 0 3 | 对于2x2矩阵，其行列式的计算公式是： 行列式 = (左上角的数 × 右下角的数) - (右上角的数 × 左下角的数) 代入给定的矩阵值，我们可以得到： 行列式 = (0 × 3) - (-4 × 0) 计算后，行列式 = 0 所以，这个矩阵的行列式是0。 |
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| zhipu | To find the determinant of a 2x2 matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, we use the formula: \[ \text{det} = ad - bc \] For the given matrix $\begin{pmatrix} 0 & -4 \\ 0 & 3 \end{pmatrix}$, we identify the elements as: - \( a = 0 \) - \( b = -4 \) - \( c = 0 \) - \( d = 3 \) Plugging these values into the determinant formula: \[ \text{det} = (0)(3) - (0)(-4) \] \[ \text{det} = 0 - 0 \] \[ \text{det} = 0 \] Therefore, the determinant of the matrix is $\boxed{0}$. |
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| deepseek-v3 | To find the determinant of the given \( 2 \times 2 \) matrix: \[ \left( \begin{array}{cc} 0 & -4 \\ 0 & 3 \\ \end{array} \right) \] we can use the formula for the determinant of a \( 2 \times 2 \) matrix: \[ \text{det}(A) = ad - bc \] where the matrix \( A \) is: \[ A = \left( \begin{array}{cc} a & b \\ c & d \\ \end{array} \right) \] **Step-by-Step Calculation:** 1. **Identify the elements of the matrix:** \[ a = 0, \quad b = -4, \quad c = 0, \quad d = 3 \] 2. **Apply the determinant formula:** \[ \text{det}(A) = (a \times d) - (b \times c) \] \[ \text{det}(A) = (0 \times 3) - (-4 \times 0) \] 3. **Perform the multiplication:** \[ \text{det}(A) = 0 - 0 \] 4. **Calculate the final result:** \[ \text{det}(A) = 0 \] **Conclusion:** The determinant of the given matrix is: \[ \boxed{0} \] |
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