| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To find the determinant of the matrix \[ \left( \begin{array}{cc} 1 & -3 \\ -3 & 4 \\ \end{array} \right), \] 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: - \(a = 1\), - \(b = -3\), - \(c = -3\), - \(d = 4\). Substituting these values into the determinant formula, we get: \[ \text{det} = (1 \cdot 4) - (-3 \cdot -3). \] Calculating the products: \[ 1 \cdot 4 = 4, \] \[ -3 \cdot -3 = 9. \] Now, subtracting these results: \[ 4 - 9 = -5. \] Thus, the determinant of the matrix is: \[ \boxed{-5} \] |
-5 |
| deepseek-r1 | To find the determinant of the matrix \[ \left( \begin{array}{cc} 1 & -3 \\ -3 & 4 \\ \end{array} \right), \] we use the formula for the determinant of a 2x2 matrix \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\), which is \(ad - bc\). Here, \(a = 1\), \(b = -3\), \(c = -3\), and \(d = 4\). Applying the formula: \[ \text{det} = (1 \cdot 4) - (-3 \cdot -3) = 4 - 9 = -5. \] To verify, we can perform row operations to convert the matrix into an upper triangular form. Adding 3 times the first row to the second row: \[ \begin{pmatrix} 1 & -3 \\ 0 & -5 \\ \end{pmatrix}. \] The determinant of this upper triangular matrix is the product of the diagonal entries: \(1 \cdot -5 = -5\). Since row operations of this type do not change the determinant, the original matrix also has determinant \(-5\). Additionally, verifying through the inverse matrix confirms the determinant. The inverse matrix, calculated using the determinant, results in the identity matrix when multiplied by the original, confirming the determinant is correct. Thus, the determinant of the matrix is \(\boxed{-5}\). |
-5 |
| gpto1 | To find the determinant of the given 2×2 matrix: \[ \begin{pmatrix} 1 & -3 \\ -3 & 4 \\ \end{pmatrix} \] We use the formula for the determinant of a 2×2 matrix: \[ \det\begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} = ad - bc \] Plugging in the values from the matrix: \[ \det = (1)(4) - (-3)(-3) = 4 - 9 = -5 \] \(\boxed{-5}\) |
-5 |
| 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 \[ \begin{pmatrix} 1 & -3 \\ -3 & 4 \end{pmatrix} \] we have \(a = 1\), \(b = -3\), \(c = -3\), and \(d = 4\). Substituting these values into the formula, we get: \[ \text{det} = (1)(4) - (-3)(-3) \] Calculating each term, we find: \[ 1 \times 4 = 4 \] and \[ (-3) \times (-3) = 9 \] Thus, the determinant is: \[ 4 - 9 = -5 \] Therefore, the determinant of the matrix is \(\boxed{-5}\). |
-5 |
| DeepSeek-Coder-V2-Lite-Instruct | To find the determinant of the given 2x2 matrix \(\left(\begin{array}{cc} 1 & -3 \\ -3 & 4 \end{array}\right)\), 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 \(\left(\begin{array}{cc} 1 & -3 \\ -3 & 4 \end{array}\right)\), we identify \(a = 1\), \(b = -3\), \(c = -3\), and \(d = 4\). Plugging these values into the determinant formula, we get: \[ \text{det} = (1)(4) - (-3)(-3) \] Calculating the products: \[ \text{det} = 4 - 9 \] Subtracting the results: \[ \text{det} = -5 \] Thus, the determinant of the matrix is: \[ \boxed{-5} \] |
-5 |
| 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 $\left(\begin{array}{cc} 1 & -3 \\ -3 & 4 \\ \end{array}\right)$ as $(1)(4) - (-3)(-3)$. ## Step 3: Perform the arithmetic $(1)(4) = 4$ and $(-3)(-3) = 9$. Therefore, the determinant is $4 - 9$. ## Step 4: Calculate the final result $4 - 9 = -5$. The final answer is: $\boxed{-5}$ |
-5 |
| 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: $(1)(4) - (-3)(-3) = 4 - 9$. ## Step 3: Simplify the expression Simplifying the expression, we get $4 - 9 = -5$. The final answer is: $\boxed{-5}$ |
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| Qwen2-72B-Instruct | To find the determinant of a 2x2 matrix \[ \left( \begin{array}{cc} a & b \\ c & d \\ \end{array} \right) \] you use the formula \[ \text{det} = ad - bc. \] For the given matrix \[ \left( \begin{array}{cc} 1 & -3 \\ -3 & 4 \\ \end{array} \right), \] we have \(a = 1\), \(b = -3\), \(c = -3\), and \(d = 4\). Plugging these values into the formula gives us: \[ \text{det} = (1)(4) - (-3)(-3) = 4 - 9 = -5. \] Therefore, the determinant of the given matrix is \(-5\). \[ \boxed{-5} \] |
-5 |
| baichuan4 | To find the determinant of a $2 \times 2$ matrix, we can use the following formula: For a matrix $$ A = \left( \begin{array}{cc} a & b \\ c & d \\ \end{array} \right), $$ the determinant is given by $$ \text{det}(A) = ad - bc. $$ Now, let's apply this formula to the given matrix: $$ \left( \begin{array}{cc} 1 & -3 \\ -3 & 4 \\ \end{array} \right). $$ Here, $a = 1$, $b = -3$, $c = -3$, and $d = 4$. So, the determinant is: $$ \text{det}(A) = (1)(4) - (-3)(-3) = 4 - 9 = -5. $$ Therefore, the determinant of the given matrix is $-5$. In a box, the answer is $$ $$ \boxed{-5}. $$ |
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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 $\begin{pmatrix}1 & -3\\-3 & 4\end{pmatrix}$, the determinant is $1\times4-(-3)\times(-3)=\boxed{4-9=-5}$. |
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| gpto1 | To find the determinant of the given 2×2 matrix: \[ \begin{pmatrix} 1 & -3 \\ -3 & 4 \\ \end{pmatrix} \] We use the formula for the determinant of a 2×2 matrix: \[ \det\begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} = ad - bc \] Plugging in the values from the matrix: \[ \det = (1)(4) - (-3)(-3) = 4 - 9 = -5 \] \(\boxed{-5}\) |
-5 |
| 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: \[ \left(\begin{array}{cc} 1 & -3 \\ -3 & 4 \\ \end{array}\right) \] We identify the elements as follows: - \(a = 1\) - \(b = -3\) - \(c = -3\) - \(d = 4\) Now, we substitute these values into the determinant formula: \[ \text{det}\left(\begin{array}{cc} 1 & -3 \\ -3 & 4 \\ \end{array}\right) = (1)(4) - (-3)(-3) \] First, calculate \(ad\): \[ (1)(4) = 4 \] Next, calculate \(bc\): \[ (-3)(-3) = 9 \] Now, subtract \(bc\) from \(ad\): \[ 4 - 9 = -5 \] Therefore, the determinant of the matrix is: \[ \boxed{-5} \] |
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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: $$\left(\begin{array}{cc} 1 & -3 \\ -3 & 4 \\ \end{array}\right)$$ The determinant is: $$\text{det} = (1)(4) - (-3)(-3) = 4 - 9 = -5$$ So the determinant of the matrix is $-5$. Therefore, the final answer in a box is: $$\boxed{-5}$$ |
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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: $\left( \begin{array}{cc} 1 & -3 \\ -3 & 4 \\ \end{array} \right)$ a = 1, b = -3, c = -3, and d = 4. So, the determinant is: $(1 \times 4) - (-3 \times -3) = 4 - 9 = -5$ Thus, the determinant of the given matrix is $\boxed{-5}$. |
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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 $\begin{pmatrix} 1 & -3 \\ -3 & 4 \end{pmatrix}$, we have: - $a = 1$ - $b = -3$ - $c = -3$ - $d = 4$ Plugging these values into the formula: \[ \text{det} = (1)(4) - (-3)(-3) \] \[ \text{det} = 4 - 9 \] \[ \text{det} = -5 \] So, the determinant of the matrix is $\boxed{-5}$. |
-5 |
| deepseek-v3 | To find the determinant of the given \( 2 \times 2 \) matrix: \[ \left( \begin{array}{cc} 1 & -3 \\ -3 & 4 \\ \end{array} \right) \] we use the formula for the determinant of a \( 2 \times 2 \) matrix: \[ \text{det}(A) = ad - bc \] where the matrix \( A \) is: \[ A = \left( \begin{array}{cc} a & b \\ c & d \\ \end{array} \right) \] **Step-by-Step Calculation:** 1. **Identify the elements of the matrix:** \[ a = 1, \quad b = -3, \quad c = -3, \quad d = 4 \] 2. **Apply the determinant formula:** \[ \text{det}(A) = (1)(4) - (-3)(-3) \] 3. **Calculate the products:** \[ \text{det}(A) = 4 - 9 \] 4. **Subtract to find the determinant:** \[ \text{det}(A) = -5 \] **Final Answer:** \[ \boxed{-5} \] |
-5 |