Math Problem Statement

show me how to solve this in detail using the Sarrus rule

We wish to calculate the determinant of a 4x4 matrix M using Sarrus' rule and compare it to two smaller matrices.

Solution

The problem involves calculating the determinant of a 4x4 matrix, and you also want to know if the given matrices in the image are equivalent in terms of their determinants.

To address this problem:

  1. Matrix MM: The 4x4 matrix is given as:
5 & 4 & -2 & 3 \\ 5 & 7 & -1 & 8 \\ 5 & 7 & 6 & 10 \\ 5 & 7 & 1 & 9 \end{pmatrix}$$ Since this is a 4x4 matrix, **Sarrus' Rule** applies only for 3x3 matrices. Therefore, we cannot directly apply the Sarrus rule to this matrix. Instead, for a 4x4 matrix, we use cofactor expansion (Laplace's Expansion), breaking it down into smaller 3x3 matrices. 2. **Smaller Matrices**: The two other determinants given involve smaller 3x3 matrices where Sarrus’ Rule **does** apply. #### Applying Sarrus' Rule to the first 3x3 matrix: $$\begin{pmatrix} 3 & 1 & 5 \\ 5 & 3 & 7 \\ 3 & 3 & 6 \end{pmatrix}$$ According to Sarrus' Rule for a 3x3 matrix, the determinant is calculated as: $$\text{Det} = aei + bfg + cdh - ceg - bdi - afh$$ Where: $$\begin{aligned} a & = 3, \, b = 1, \, c = 5, \\ d & = 5, \, e = 3, \, f = 7, \\ g & = 3, \, h = 3, \, i = 6 \end{aligned}$$ Plugging in values: $$\text{Det} = (3 \times 3 \times 6) + (1 \times 7 \times 3) + (5 \times 5 \times 3) - (5 \times 3 \times 3) - (1 \times 5 \times 6) - (3 \times 7 \times 3)$$ Let's calculate each term: $$\text{Det} = (54) + (21) + (75) - (45) - (30) - (63)$$ $$\text{Det} = 150 - 138 = 12$$ So, the determinant of the 3x3 matrix is **12**. #### Applying Sarrus' Rule to the second 3x3 matrix: $$\begin{pmatrix} 7 & 2 \\ 2 & 1 \end{pmatrix}$$ This is a simple 2x2 matrix determinant: $$\text{Det} = (7 \times 1) - (2 \times 2) = 7 - 4 = 3$$ The 3x3 version would involve scaling by 5, as indicated. So the determinant is **5 x 3 = 15**. #### Equivalence: 1. **First 3x3 matrix** has determinant 12. 2. **Second matrix** (5 × 2x2) results in a determinant of 15. These results are **not equivalent**. Would you like further clarification on these calculations? ### Related Questions: 1. How do we expand a 4x4 determinant using cofactor expansion? 2. Can the Sarrus Rule be applied to matrices larger than 3x3? 3. What other methods can be used to calculate determinants of 4x4 matrices? 4. What is the geometric interpretation of the determinant? 5. How does row or column operations affect the determinant of a matrix? **Tip:** Sarrus’ Rule is a quick way to calculate determinants but is limited to 3x3 matrices. Larger matrices require other techniques like cofactor expansion or reduction to row echelon form.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Determinants

Formulas

Determinant of 3x3 matrix: Det = aei + bfg + cdh - ceg - bdi - afh
Determinant of 2x2 matrix: Det = ad - bc

Theorems

Sarrus' Rule
Cofactor Expansion

Suitable Grade Level

Undergraduate

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