| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | The determinant of a 3x3 matrix A = [[a, b, c], [d, e, f], [g, h, i]] can be calculated using the formula: det(A) = a*(e*i - f*h) - b*(d*i - f*g) + c*(d*h - e*g) For the given matrix A = [[2, 4, 3], [3, 3, 1], [42, 20, 51]], we can substitute the values into the formula: det(A) = 2*(3*51 - 1*20) - 4*(3*51 - 1*42) + 3*(3*20 - 3*42) = 2*(153 - 20) - 4*(153 - 42) + 3*(60 - 126) = 2*133 - 4*111 + 3*(-66) = 266 - 444 - 198 = -376 Therefore, the determinant of matrix A is -376. |
-376 |
| deepseekr1 | The determinant of matrix \( A = \begin{bmatrix} 2 & 4 & 3 \\ 3 & 3 & 1 \\ 42 & 20 & 51 \end{bmatrix} \) is calculated using the cofactor expansion along the first row: \[ \begin{align*} \text{det}(A) &= 2 \cdot \begin{vmatrix} 3 & 1 \\ 20 & 51 \end{vmatrix} - 4 \cdot \begin{vmatrix} 3 & 1 \\ 42 & 51 \end{vmatrix} + 3 \cdot \begin{vmatrix} 3 & 3 \\ 42 & 20 \end{vmatrix} \\ &= 2 \cdot (3 \cdot 51 - 1 \cdot 20) - 4 \cdot (3 \cdot 51 - 1 \cdot 42) + 3 \cdot (3 \cdot 20 - 3 \cdot 42) \\ &= 2 \cdot (153 - 20) - 4 \cdot (153 - 42) + 3 \cdot (60 - 126) \\ &= 2 \cdot 133 - 4 \cdot 111 + 3 \cdot (-66) \\ &= 266 - 444 - 198 \\ &= -376. \end{align*} \] **Answer:** The determinant is \(\boxed{-376}\). |
-376 |
| gpto1 | To find the determinant of the given 3×3 matrix: \[ A = \begin{bmatrix} 2 & 4 & 3 \\ 3 & 3 & 1 \\ 42 & 20 & 51 \end{bmatrix} \] We can use the formula for the determinant of a 3×3 matrix: \[ \det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \] where the matrix elements are: \[ \begin{matrix} a = 2, & b = 4, & c = 3 \\ d = 3, & e = 3, & f = 1 \\ g = 42, & h = 20, & i = 51 \end{matrix} \] **Step 1: Calculate the three minor determinants.** 1. **Compute \( ei - fh \):** \[ ei = e \times i = 3 \times 51 = 153 \\ fh = f \times h = 1 \times 20 = 20 \\ ei - fh = 153 - 20 = 133 \] 2. **Compute \( di - fg \):** \[ di = d \times i = 3 \times 51 = 153 \\ fg = f \times g = 1 \times 42 = 42 \\ di - fg = 153 - 42 = 111 \] 3. **Compute \( dh - eg \):** \[ dh = d \times h = 3 \times 20 = 60 \\ eg = e \times g = 3 \times 42 = 126 \\ dh - eg = 60 - 126 = -66 \] **Step 2: Apply the determinant formula.** \[ \det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \] Plug in the calculated values: \[ \det(A) = 2 \times 133 - 4 \times 111 + 3 \times (-66) \] **Step 3: Compute each term.** 1. First term: \[ 2 \times 133 = 266 \] 2. Second term: \[ -4 \times 111 = -444 \] 3. Third term: \[ 3 \times (-66) = -198 \] **Step 4: Sum the terms to find the determinant.** \[ \det(A) = 266 - 444 - 198 = (266 - 444) - 198 = (-178) - 198 = -376 \] **Answer:** The determinant of the matrix \( A \) is **-376**. |
-376 |
| GPT-4o | To find the determinant of a 3x3 matrix \( A \), we can use the formula for the determinant of a 3x3 matrix. Given the matrix \( A \): \[ A = \begin{pmatrix} 2 & 4 & 3 \\ 3 & 3 & 1 \\ 42 & 20 & 51 \end{pmatrix} \] The determinant of \( A \), denoted as \( \det(A) \), can be calculated using the rule for 3x3 matrices: \[ \det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \] where the matrix elements are: \[ A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \] In this case: \[ a = 2, \, b = 4, \, c = 3, \, d = 3, \, e = 3, \, f = 1, \, g = 42, \, h = 20, \, i = 51 \] Now, substitute these values into the determinant formula: \[ \det(A) = 2(3 \cdot 51 - 1 \cdot 20) - 4(3 \cdot 51 - 1 \cdot 42) + 3(3 \cdot 20 - 3 \cdot 42) \] Calculate each term separately: \[ 2(3 \cdot 51 - 1 \cdot 20) = 2(153 - 20) = 2 \cdot 133 = 266 \] \[ 4(3 \cdot 51 - 1 \cdot 42) = 4(153 - 42) = 4 \cdot 111 = 444 \] \[ 3(3 \cdot 20 - 3 \cdot 42) = 3(60 - 126) = 3 \cdot (-66) = -198 \] Combine these results: \[ \det(A) = 266 - 444 - 198 \] \[ \det(A) = 266 - 444 - 198 = -176 - 198 = -374 \] So, the determinant of matrix \( A \) is: \[ \det(A) = -374 \] |
-376 |
| llama3.1-70B-Instruct | To find the determinant of matrix A, we can use the formula for the determinant of a 3x3 matrix: det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) where a, b, c, d, e, f, g, h, and i are the elements of the matrix. For matrix A = [[2, 4, 3], [3, 3, 1], [42, 20, 51]], we have: a = 2, b = 4, c = 3 d = 3, e = 3, f = 1 g = 42, h = 20, i = 51 Plugging these values into the formula, we get: det(A) = 2(3*51 - 1*20) - 4(3*51 - 1*42) + 3(3*20 - 3*42) = 2(153 - 20) - 4(153 - 42) + 3(60 - 126) = 2(133) - 4(111) + 3(-66) = 266 - 444 - 198 = -376 Therefore, the determinant of matrix A is -376. |
-376 |
| llama3.1-405B-Instruct | import numpy as np A = np.array([[2, 4, 3], [3, 3, 1], [42, 20, 51]]) det_A = np.linalg.det(A) print("The determinant of matrix A is: ", det_A) |
-376 |
| DeepSeek-Coder-V2 | To find the determinant of matrix A = [[2, 4, 3], [3, 3, 1], [42, 20, 51]], we can use the formula for the determinant of a 3x3 matrix: det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) where the elements of the matrix are: a = 2, b = 4, c = 3 d = 3, e = 3, f = 1 g = 42, h = 20, i = 51 Now, let's calculate each part of the formula: ei - fh = (3 * 51) - (1 * 20) = 153 - 20 = 133 di - fg = (3 * 51) - (1 * 42) = 153 - 42 = 111 dh - eg = (3 * 20) - (3 * 42) = 60 - 126 = -66 Now, substitute these values back into the determinant formula: det(A) = 2(133) - 4(111) + 3(-66) det(A) = 266 - 444 - 198 det(A) = 266 - 642 det(A) = -376 So, the determinant of matrix A is -376. |
-376 |
| Qwen2-72B-Instruct | The determinant of a 3x3 matrix A = [[a, b, c], [d, e, f], [g, h, i]] can be calculated using the formula: det(A) = a*(e*i - f*h) - b*(d*i - f*g) + c*(d*h - e*g) For the given matrix A = [[2, 4, 3], [3, 3, 1], [42, 20, 51]], we can substitute the values into the formula: det(A) = 2*(3*51 - 1*20) - 4*(3*51 - 1*42) + 3*(3*20 - 3*42) = 2*(153 - 20) - 4*(153 - 42) + 3*(60 - 126) = 2*133 - 4*111 + 3*(-66) = 266 - 444 - 198 = -376 Therefore, the determinant of matrix A is -376. |
-376 |
| baichuan4 | To find the determinant of matrix A = [[2, 4, 3], [3, 3, 1], [42, 20, 51]], we can use the rule of Sarrus or apply row reduction to convert the matrix into a triangular form and then calculate the determinant as the product of the diagonal elements. Let's use the rule of Sarrus: ``` |A| = 2(3*51 - 1*20) - 4(3*42 - 1*3) + 3(3*20 - 42*1) = 2(153 - 20) - 4(126 - 3) + 3(60 - 42) = 2(133) - 4(123) + 3(18) = 266 - 492 + 54 = -172 + 54 = -118 ``` So, the determinant of matrix A is -118. |
-376 |
| doubao | To find the determinant of a 3x3 matrix [[a,b,c],[d,e,f],[g,h,i]], the formula is: det = a(ei - fh) - b(di - fg) + c(dh - eg). For the given matrix A = [[2,4,3],[3,3,1],[42,20,51]], the determinant is: det = 2((3*51) - (1*20)) - 4((3*51) - (1*42)) + 3((3*20) - (3*42)) = 2(153 - 20) - 4(153 - 42) + 3(60 - 126) = 2*133 - 4*111 + 3*-66 = 266 - 444 - 198 = -376. |
-376 |
| minimax | To find the determinant of a 3x3 matrix A, we can use the following formula: det(A) = a11(a22a33 - a23a32) - a12(a21a33 - a23a31) + a13(a21a32 - a22a31) where aij represents the element in the i-th row and j-th column of matrix A. Given matrix A: A = [[2, 4, 3], [3, 3, 1], [42, 20, 51]] We can calculate its determinant as follows: det(A) = 2(3*51 - 1*20) - 4(3*51 - 1*42) + 3(3*20 - 3*42) = 2(153 - 20) - 4(153 - 42) + 3(60 - 126) = 2(133) - 4(111) + 3(-66) = 266 - 444 - 198 = -178 - 198 = -376 So, the determinant of matrix A is -376. |
-376 |
| kimi | To find the determinant of a 3x3 matrix A, you can use the following formula: For a matrix A = [[a, b, c], [d, e, f], [g, h, i]], the determinant det(A) is calculated as: det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) Given matrix A = [[2, 4, 3], [3, 3, 1], [42, 20, 51]], let's assign the values to a, b, c, d, e, f, g, h, and i: a = 2, b = 4, c = 3 d = 3, e = 3, f = 1 g = 42, h = 20, i = 51 Now, apply the formula: det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) = 2((3*51) - (1*20)) - 4((3*42) - (1*42)) + 3((3*20) - (3*42)) = 2(153 - 20) - 4(126 - 42) + 3(60 - 126) = 2(133) - 4(84) + 3(-66) = 266 - 336 - 198 = -370 So, the determinant of matrix A is -370. |
-376 |
| wenxinyiyan | To find the determinant of a 3x3 matrix like A, we use the formula: det(A) = a[0][0] * (a[1][1]*a[2][2] - a[2][1]*a[1][2]) - a[0][1] * (a[1][0]*a[2][2] - a[1][2]*a[2][0]) + a[0][2] * (a[1][0]*a[2][1] - a[1][1]*a[2][0]) where a[i][j] represents the element in the ith row and jth column of matrix A. For the given matrix A = [[2, 4, 3], [3, 3, 1], [42, 20, 51]], let's apply the formula: det(A) = 2 * (3*51 - 1*20) - 4 * (3*51 - 1*42) + 3 * (3*20 - 3*42) = 2 * (153 - 20) - 4 * (153 - 42) + 3 * (60 - 126) = 2 * 133 - 4 * 111 + 3 * (-66) = 266 - 444 - 198 = -376 So, the determinant of matrix A is -376. |
-376 |
| zhipu | To find the determinant of a 3x3 matrix, you can use the rule for the determinant of a 3x3 matrix which is given by: For a matrix A: ``` A = | a b c | | d e f | | g h i | ``` The determinant (det(A)) is calculated as: ``` det(A) = aei + bfg + cdh - ceg - bdi - afh ``` For the given matrix A: ``` A = [[2, 4, 3], [3, 3, 1], [42, 20, 51]] ``` We have: ``` a = 2, b = 4, c = 3 d = 3, e = 3, f = 1 g = 42, h = 20, i = 51 ``` Plugging these values into the formula for the determinant, we get: ``` det(A) = (2)(3)(51) + (4)(1)(42) + (3)(3)(20) - (3)(3)(42) - (4)(3)(51) - (2)(1)(20) ``` Calculating each term separately: ``` (2)(3)(51) = 306 (4)(1)(42) = 168 (3)(3)(20) = 180 (3)(3)(42) = 378 (4)(3)(51) = 612 (2)(1)(20) = 40 ``` Now, summing the positive terms and subtracting the negative terms: ``` det(A) = 306 + 168 + 180 - 378 - 612 - 40 ``` Adding the positive terms: ``` 306 + 168 + 180 = 654 ``` Adding the negative terms: ``` 378 + 612 + 40 = 1030 ``` Finally, subtracting the negative terms sum from the positive terms sum: ``` det(A) = 654 - 1030 = -376 ``` So, the determinant of matrix A is -376. |
-376 |
| gpto1 | To find the determinant of the given 3×3 matrix: \[ A = \begin{bmatrix} 2 & 4 & 3 \\ 3 & 3 & 1 \\ 42 & 20 & 51 \end{bmatrix} \] We can use the formula for the determinant of a 3×3 matrix: \[ \det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \] where the matrix elements are: \[ \begin{matrix} a = 2, & b = 4, & c = 3 \\ d = 3, & e = 3, & f = 1 \\ g = 42, & h = 20, & i = 51 \end{matrix} \] **Step 1: Calculate the three minor determinants.** 1. **Compute \( ei - fh \):** \[ ei = e \times i = 3 \times 51 = 153 \\ fh = f \times h = 1 \times 20 = 20 \\ ei - fh = 153 - 20 = 133 \] 2. **Compute \( di - fg \):** \[ di = d \times i = 3 \times 51 = 153 \\ fg = f \times g = 1 \times 42 = 42 \\ di - fg = 153 - 42 = 111 \] 3. **Compute \( dh - eg \):** \[ dh = d \times h = 3 \times 20 = 60 \\ eg = e \times g = 3 \times 42 = 126 \\ dh - eg = 60 - 126 = -66 \] **Step 2: Apply the determinant formula.** \[ \det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \] Plug in the calculated values: \[ \det(A) = 2 \times 133 - 4 \times 111 + 3 \times (-66) \] **Step 3: Compute each term.** 1. First term: \[ 2 \times 133 = 266 \] 2. Second term: \[ -4 \times 111 = -444 \] 3. Third term: \[ 3 \times (-66) = -198 \] **Step 4: Sum the terms to find the determinant.** \[ \det(A) = 266 - 444 - 198 = (266 - 444) - 198 = (-178) - 198 = -376 \] **Answer:** The determinant of the matrix \( A \) is **-376**. |
-376 |
| deepseekv3 | To calculate the determinant of matrix \( A \), we can use the rule for a \( 3 \times 3 \) matrix. Given: \[ A = \begin{bmatrix} 2 & 4 & 3 \\ 3 & 3 & 1 \\ 42 & 20 & 51 \end{bmatrix} \] The determinant of \( A \), denoted \( \det(A) \), is calculated as: \[ \det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \] where the elements of the matrix are labeled as: \[ \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \] Substituting the values from matrix \( A \): \[ a = 2, \quad b = 4, \quad c = 3 \] \[ d = 3, \quad e = 3, \quad f = 1 \] \[ g = 42, \quad h = 20, \quad i = 51 \] Now, compute each part: 1. \( ei - fh = (3)(51) - (1)(20) = 153 - 20 = 133 \) 2. \( di - fg = (3)(51) - (1)(42) = 153 - 42 = 111 \) 3. \( dh - eg = (3)(20) - (3)(42) = 60 - 126 = -66 \) Substitute these into the determinant formula: \[ \det(A) = 2(133) - 4(111) + 3(-66) \] \[ \det(A) = 266 - 444 - 198 \] \[ \det(A) = 266 - 444 = -178 \] \[ \det(A) = -178 - 198 = -376 \] Thus, the determinant of matrix \( A \) is: \[ \boxed{-376} \] |
-376 |