| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To find the determinant of the matrix \[ \left( \begin{array}{cc} 3 & 5 \\ -4 & -4 \\ \end{array} \right), \] 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: - \(a = 3\), - \(b = 5\), - \(c = -4\), - \(d = -4\). Substituting these values into the determinant formula, we get: \[ \text{det} = (3 \cdot -4) - (5 \cdot -4). \] Calculating each term separately: \[ 3 \cdot -4 = -12, \] \[ 5 \cdot -4 = -20. \] Now, substituting these results back into the formula: \[ \text{det} = -12 - (-20). \] Since subtracting a negative is the same as adding the positive: \[ \text{det} = -12 + 20 = 8. \] Thus, the determinant of the matrix is: \[ \boxed{8} \] |
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| deepseek-r1 | To find the determinant of the matrix \[ \left( \begin{array}{cc} 3 & 5 \\ -4 & -4 \\ \end{array} \right), \] we use the formula for the determinant of a 2x2 matrix \(\left( \begin{array}{cc} a & b \\ c & d \end{array} \right)\), which is \(ad - bc\). For the given matrix, the entries are \(a = 3\), \(b = 5\), \(c = -4\), and \(d = -4\). Applying the formula: \[ \text{det} = (3)(-4) - (5)(-4). \] Calculating each term: \[ (3)(-4) = -12, \] \[ (5)(-4) = -20. \] Subtracting these products: \[ -12 - (-20) = -12 + 20 = 8. \] Thus, the determinant of the matrix is \(\boxed{8}\). |
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| gpto1 | To find the determinant of the matrix: \[ \left( \begin{array}{cc} 3 & 5 \\ -4 & -4 \\ \end{array} \right) \] We use the formula for the determinant of a 2×2 matrix: \[ \text{determinant} = (a)(d) - (b)(c) \] where \( a, b, c, d \) are the entries of the matrix: \[ a = 3,\quad b = 5,\quad c = -4,\quad d = -4 \] Plugging in the values: \[ \text{determinant} = (3)(-4) - (5)(-4) = -12 - (-20) = -12 + 20 = 8 \] So the determinant is 8. \(\boxed{8}\) |
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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 \[ \begin{pmatrix} 3 & 5 \\ -4 & -4 \end{pmatrix} \] we have \(a = 3\), \(b = 5\), \(c = -4\), and \(d = -4\). Substituting these values into the determinant formula, we get: \[ \text{det} = (3)(-4) - (5)(-4) \] Calculating each term separately: \[ (3)(-4) = -12 \] \[ (5)(-4) = -20 \] Now, substitute these results back into the determinant expression: \[ \text{det} = -12 - (-20) = -12 + 20 = 8 \] Thus, the determinant of the matrix is \(\boxed{8}\). |
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| DeepSeek-Coder-V2-Lite-Instruct | To find the determinant of the given 2x2 matrix \(\left(\begin{array}{cc} 3 & 5 \\ -4 & -4 \end{array}\right)\), we use the formula for the determinant of a 2x2 matrix \(\left(\begin{array}{cc} a & b \\ c & d \end{array}\right)\), which is given by: \[ \text{det} = ad - bc \] For the matrix \(\left(\begin{array}{cc} 3 & 5 \\ -4 & -4 \end{array}\right)\), we identify \(a = 3\), \(b = 5\), \(c = -4\), and \(d = -4\). Plugging these values into the determinant formula, we get: \[ \text{det} = (3 \cdot -4) - (5 \cdot -4) \] Calculating the products: \[ 3 \cdot -4 = -12 \] \[ 5 \cdot -4 = -20 \] Now, subtract the second product from the first: \[ -12 - (-20) = -12 + 20 = 8 \] Thus, the determinant of the matrix is: \[ \boxed{8} \] |
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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 Using the formula, we can calculate the determinant of the given matrix $\left(\begin{array}{cc} 3 & 5 \\ -4 & -4 \\ \end{array}\right)$ as $(3)(-4) - (5)(-4)$. ## Step 3: Perform the arithmetic $(3)(-4) = -12$ and $(5)(-4) = -20$. Therefore, the determinant is $-12 - (-20)$. ## Step 4: Simplify the expression $-12 - (-20) = -12 + 20 = 8$. The final answer is: $\boxed{8}$ |
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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 $\left(\begin{array}{cc}a & b \\ c & d\end{array}\right)$ is given by the formula $ad - bc$. ## Step 2: Apply the formula to the given matrix Using the formula, we can calculate the determinant of the given matrix as follows: $(3)(-4) - (5)(-4)$. ## Step 3: Simplify the expression $(3)(-4) - (5)(-4) = -12 + 20$. ## Step 4: Calculate the final result $-12 + 20 = 8$. The final answer is: $\boxed{8}$ |
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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 $\left(\begin{array}{cc}3 & 5 \\-4 & -4 \\\end{array}\right)$, we have: $a = 3$, $b = 5$, $c = -4$, and $d = -4$. Thus, the determinant is: $3(-4) - 5(-4) = -12 + 20 = 8$. Therefore, the determinant of the given matrix is $\boxed{8}$. |
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| baichuan4 | To find the determinant of a $2 \times 2$ matrix, we use the formula: $$\text{det}\left(\begin{array}{cc} a & b \\ c & d \\ \end{array}\right) = ad - bc$$ Given the matrix: $$\left(\begin{array}{cc} 3 & 5 \\ -4 & -4 \\ \end{array}\right)$$ We can identify the elements as follows: - $a = 3$ - $b = 5$ - $c = -4$ - $d = -4$ Now, we apply the formula to calculate the determinant: $$\text{det} = (3)(-4) - (5)(-4)$$ $$\text{det} = -12 + 20$$ $$\text{det} = 8$$ Therefore, the determinant of the given matrix is $8$. We can present this answer in a boxed format as requested: $$\boxed{8}$$ |
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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 $\begin{pmatrix}3 & 5\\-4 & -4\end{pmatrix}$, the determinant is $3\times(-4)-5\times(-4)=-12+20=8$. So the answer is $\boxed{8}$. |
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| gpto1 | To find the determinant of the matrix: \[ \left( \begin{array}{cc} 3 & 5 \\ -4 & -4 \\ \end{array} \right) \] We use the formula for the determinant of a 2×2 matrix: \[ \text{determinant} = (a)(d) - (b)(c) \] where \( a, b, c, d \) are the entries of the matrix: \[ a = 3,\quad b = 5,\quad c = -4,\quad d = -4 \] Plugging in the values: \[ \text{determinant} = (3)(-4) - (5)(-4) = -12 - (-20) = -12 + 20 = 8 \] So the determinant is 8. \(\boxed{8}\) |
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| kimi | To find the determinant of the given 2x2 matrix, we will use the formula for the determinant of a 2x2 matrix: \[ \text{det}\left(\begin{array}{cc} a & b \\ c & d \\ \end{array}\right) = ad - bc \] Given the matrix: \[ \left(\begin{array}{cc} 3 & 5 \\ -4 & -4 \\ \end{array}\right) \] We identify the elements as follows: - \(a = 3\) - \(b = 5\) - \(c = -4\) - \(d = -4\) Now, we substitute these values into the determinant formula: \[ \text{det}\left(\begin{array}{cc} 3 & 5 \\ -4 & -4 \\ \end{array}\right) = (3)(-4) - (5)(-4) \] Next, we perform the multiplications: \[ (3)(-4) = -12 \] \[ (5)(-4) = -20 \] Then, we subtract the second product from the first: \[ -12 - (-20) = -12 + 20 = 8 \] Therefore, the determinant of the matrix is: \[ \boxed{8} \] |
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| minimax | To find the determinant of a 2x2 matrix, you use the formula: $$\text{det}\left(\begin{array}{cc} a & b \\ c & d \\ \end{array}\right) = ad - bc$$ For the given matrix: $$\left(\begin{array}{cc} 3 & 5 \\ -4 & -4 \\ \end{array}\right)$$ The determinant is: $$\text{det} = (3)(-4) - (5)(-4) = -12 + 20 = 8$$ So the determinant of the matrix is $\boxed{8}$. |
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| wenxinyiyan | To find the determinant of a 2x2 matrix, we use the formula: $\text{Determinant} = (a \times d) - (b \times c)$ For the given matrix: $\left( \begin{array}{cc} 3 & 5 \\ -4 & -4 \\ \end{array} \right)$ We have $a = 3$, $b = 5$, $c = -4$, and $d = -4$. Plugging these values into the formula, we get: $\text{Determinant} = (3 \times -4) - (5 \times -4)$ $= -12 + 20$ $= 8$ So, the determinant of the given matrix is $\boxed{8}$. |
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| zhipu | To find the determinant of a \(2 \times 2\) matrix \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\), we use the formula: \[ \text{det} = ad - bc \] Given the matrix \(\begin{pmatrix} 3 & 5 \\ -4 & -4 \end{pmatrix}\), we identify \(a = 3\), \(b = 5\), \(c = -4\), and \(d = -4\). Now, substitute these values into the determinant formula: \[ \text{det} = (3)(-4) - (5)(-4) \] Calculate each term: \[ (3)(-4) = -12 \] \[ (5)(-4) = -20 \] Now, compute the determinant: \[ \text{det} = -12 - (-20) = -12 + 20 = 8 \] Thus, the determinant of the matrix is \(\boxed{8}\). |
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| deepseek-v3 | To find the determinant of the \( 2 \times 2 \) matrix: \[ \left( \begin{array}{cc} 3 & 5 \\ -4 & -4 \\ \end{array} \right) \] we use the formula for the determinant of a \( 2 \times 2 \) matrix: \[ \text{det}(A) = ad - bc \] where the matrix \( A \) is: \[ A = \left( \begin{array}{cc} a & b \\ c & d \\ \end{array} \right) \] Applying the values from the given matrix: \[ a = 3, \quad b = 5, \quad c = -4, \quad d = -4 \] Now, calculate the determinant: \[ \text{det}(A) = (3)(-4) - (5)(-4) \] \[ \text{det}(A) = -12 - (-20) \] \[ \text{det}(A) = -12 + 20 \] \[ \text{det}(A) = 8 \] \[ \boxed{8} \] |
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