Problem 46 - Precalculus: Functions and Graphs

Question

Find the determinant of the matrix. $$\left[\begin{array}{cccc} 4.2 & 1.7 & -2 & -4 \\ -7 & 0.1 & 4.6 & 2.7 \\ 4.1 & -7 & 12 & 6.8 \\ 4.6 & 2 & 3.2 & 1.2 \end{array}\right]$$

Step-by-Step Solution

Answer

Determinant = 49.328

Step 1: Understand the Matrix

The given matrix is a 4x4 matrix, which requires applying the formula for the determinant of a 4x4 matrix through expansion by minors.

Step 2: Choose the Row for Expansion

Choose to expand along the first row: 4.2, 1.7, -2, -4, as this row simplifies calculations.

Step 3: Calculate the Cofactor for First Element

The element is 4.2. Remove the first row and column to get the 3x3 matrix:\[\begin{vmatrix} 0.1 & 4.6 & 2.7 \ -7 & 12 & 6.8 \ 2 & 3.2 & 1.2 \end{vmatrix}\]Calculate its determinant using the 3x3 formula.

Step 4: Compute Minor for 3x3 Matrix (Element 4.2)

Compute the determinant of the 3x3 matrix obtained in Step 3 to find the minor for the 4.2 element:\[0.1(12 \cdot 1.2 - 6.8 \cdot 3.2) - 4.6(-7 \cdot 1.2 - 6.8 \cdot 2) + 2.7(-7 \cdot 3.2 - 12 \cdot 2)\].

Step 5: Repeat Steps 3-4 for Remaining Elements

Repeat the calculation of minors and cofactors for 1.7, -2, and -4, by removing their respective rows and columns and calculating their 3x3 determinants.

Step 6: Sum Up the Cofactor Expansion

Use the cofactor expansion formula for the determinant of a 4x4 matrix:\[\det(A) = 4.2 \cdot C_{11} - 1.7 \cdot C_{12} + (-2) \cdot C_{13} - 4 \cdot C_{14}\]where \(C_{ij}\) are the cofactors found in previous steps.

Step 7: Calculate the Final Determinant Value

After finding the values of each cofactor, substitute these back into the formula from Step 6 to calculate the determinant.

Key Concepts

4x4 Matrix Determinant

Determining the determinant of a 4x4 matrix can be a bit more complex compared to smaller matrices. However, it's a crucial skill to master as it helps in various applications in algebra and calculus.
Understanding the foundational approach involves breaking it down using a method called "expansion by minors."
This means we take a 4x4 matrix and expand it into several smaller 3x3 matrices.

• The determinant of each of these smaller matrices contributes to the final determinant calculation of the 4x4 matrix.

It's like solving a large puzzle by breaking it into smaller, more manageable pieces. Remember, while the size increases, the fundamental approach remains consistent.

Expansion by Minors

Expansion by minors is a method used to simplify the computation of larger matrices.
This involves choosing a row or column, then breaking down the determinant calculation into steps using smaller matrices.
For a 4x4 matrix, you'll choose a row or column—a common tactic is choosing the one with the most zeros to ease calculations.

• Each element in the row or column chosen will have an associated cofactor.

The determinant of the original matrix is expressed as the sum of these cofactors, multiplied by their corresponding elements from the chosen row or column.

Cofactor Calculation

Cofactor calculation is a critical part of finding a determinant using expansion by minors.
A cofactor is essentially a minor (determinant of a submatrix) multiplied by \(-1^{i+j}\) where \(i\) and \(j\) are the row and column indices of the matrix element.
This factor \((-1)^{i+j}\) ensures the right sign in the matrix expansion.

• To find a cofactor, remove the row and column of the element you're focusing on, leaving a smaller matrix behind.

Next, compute the determinant of this smaller matrix.
The product of this determinant and \((-1)^{i+j}\) gives you the cofactor.
Repeat for each element in the chosen row or column to gather all necessary cofactors for the determinant.

3x3 Determinant

The computation of the 3x3 determinant is an intermediate step when solving for the determinant of a 4x4 matrix.
Once a 3x3 submatrix has been identified, we use a specific formula to calculate its determinant.

• The formula involves taking the sum and difference of products based on diagonals of the matrix.

For a 3x3 matrix:\[\begin{vmatrix} a & b & c \ d & e & f \ g & h & i \end{vmatrix} = a(ei-fh) - b(di-fg) + c(dh-eg)\]Each term in this formula represents a diagonal product and is adjusted by specific arithmetic operations.
This essential step contributes the minor used in the cofactor calculation for the larger matrix determinant.