Problem 43 - College Algebra and Calculus: An Applied Approach

Question

Find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. Use a graphing utility to confirm your result. $$ \left[\begin{array}{rrr} 0.3 & 0.2 & 0.2 \\ 0.2 & 0.2 & 0.2 \\ -0.4 & 0.4 & 0.3 \end{array}\right] $$

Step-by-Step Solution

Answer

The determinant of the matrix is \(0.01\).

Step 1: Selecting the Row or Column for Expansion

We'll choose the first row to compute the determinant because it has the fewest number of computations among others, given that we can consider the non-zero elements. Our matrix is:\[\begin{bmatrix}0.3 & 0.2 & 0.2 \0.2 & 0.2 & 0.2 \-0.4 & 0.4 & 0.3\end{bmatrix}\]

Step 2: Computing the Cofactors

We now remove the first row and column to form a 2x2 sub matrix. From this sub matrix, find its determinant (found by multiplying the first element and the last element and subtracting the product of the second and third elements). We repeat this method for each element of the row, remembering to alternate the signs: cofactor of \(0.3\) is given by\[\begin{bmatrix}0.2 & 0.2 \0.4 & 0.3\end{bmatrix}\]Therefore determinant of \(0.3\) is \(0.3(0.2*0.3 - 0.2*0.4)\).cofactor of \(0.2\) is given by \[\begin{bmatrix}0.2 & 0.2 \-0.4 & 0.3\end{bmatrix}\]Therefore determinant of \(0.2\) is \(-0.2(0.2*0.3 - 0.2*(-0.4))\).cofactor of \(0.2\) is given by \[\begin{bmatrix}0.2 & 0.2 \-0.4 & 0.4\end{bmatrix}\]Therefore determinant of \(0.2\) is \(0.2(0.2*0.4 - 0.2*(-0.4))\).

Step 3: Computing the Determinant

Now add all the above determinants calculated to get the final determinant\[0.3(0.2*0.3 - 0.2*0.4) - 0.2(0.2*0.3 - 0.2*(-0.4)) + 0.2(0.2*0.4 - 0.2*(-0.4))\]

Step 4: Simplifying the Expression

Simplify the above expression to get the final answer.\[0.3*0.02 - 0.2*0.08 + 0.2*0.12 \]

Key Concepts

Cofactor Expansion

Cofactor expansion is an essential method for determining the determinant of a matrix, especially when working with larger matrices. The process involves breaking down the complex matrix into smaller ones, making calculations manageable.

Here's how cofactor expansion works:

• Choose a row or column to expand along. Usually, the row or column with the most zeros is preferred to simplify the process.
• For each element in the chosen row or column, calculate its cofactor. This involves removing the row and column of the element to form a smaller matrix, called a minor.
• The cofactor of an element is determined by multiplying its minor's determinant by the sign pattern (-1)^{i+j}, where i and j are the row and column indices.
• Sum the products of each element and its respective cofactor to find the determinant.

Cofactor expansion allows us to find determinants efficiently by leveraging these smaller computations.

2x2 Submatrix

When calculating cofactors, you'll often encounter a 2x2 submatrix. These matrices are the result of removing one row and one column from the original matrix.

Finding the determinant of a 2x2 submatrix is straightforward:

• The determinant is computed using the formula:\( \text{Det}( \begin{bmatrix} a & b \ c & d \end{bmatrix}) = ad - bc \)
• This is done by multiplying the diagonal elements and subtracting the product of the other two.

This method of dealing with 2x2 submatrices is not only simple but also crucial for successfully applying cofactor expansion.

Matrix Simplification

Matrix simplification is the final step in finding the determinant, where all previously computed cofactors are combined.

The goal is to simplify the expression to reach a numerical value. Here’s how you can do that:

• Multiply each element of the chosen row or column by its respective cofactor.
• Sum these products, taking care to respect any alternating signs.
• Perform any necessary arithmetic operations to simplify the expression.

Matrix simplification helps in arriving at the determinant value quickly and accurately, ensuring that all calculations are correctly executed.