Problem 44 - 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. $$ \left[\begin{array}{rrr} 0.1 & 0.2 & 0.3 \\ -0.3 & 0.2 & 0.2 \\ 0.5 & 0.4 & 0.4 \end{array}\right] $$

Step-by-Step Solution

Answer

The determinant of the given matrix is 0.

Step 1: Choosing Row or Column for Expansion

Since there are no zeros, we can expand along any row or column. For this solution, let's choose the first row, made up of \(0.1\), \(0.2\), and \(0.3\).

Step 2: Calculating the Cofactors

Next is to calculate the cofactor for each element in the chosen row. Let's start with the first element \(0.1\). Remove the first row and first column, which leaves the 2x2 matrix: \[\left[\begin{array}{rr} 0.2 & 0.2 \ 0.4 & 0.4 \end{array}\right]\] The determinant (cofactor) is \(0.2*0.4 - 0.2*0.4 = 0\). The process is repeated for the second and third elements of the first row.

Step 3: Multiplying Elements by their Cofactors

As per rules for determinant calculation, the original elements of the row have to be multiplied by their respective cofactors. Therefore, we compute the product for each element in the first row and their corresponding minor's determinant, then subtract the result of multiplying the second element by its corresponding minor and add the result of multiplying the third element by its minor. This gives the determinant of our 3x3 matrix.

Step 4: Final Calculation and Result Confirmation

At this point, perform the calculations taking into account the signs. The final determinant should then be calculated as \(0.1*0 - 0.2*0 + 0.3*0 = 0\). This can be confirmed using a graphing utility.

Key Concepts

Cofactor Expansion

Cofactor expansion is a method used to calculate the determinant of a matrix. It involves expanding along a row or column. Let's demystify this step by step. Think of it like opening a suitcase to see everything inside neatly packed.

• First, pick a row or column to expand. Typically, choosing one with zeros makes the math easier because calculations involving zeros simplify the process significantly.
• But don't fret; any row or column will work. In our example, we chose the first row despite no zeros.
• Next, compute each cofactor for the elements of this row or column. A cofactor for any given element is found by removing the row and column of that element, giving a smaller sub-matrix. Calculate the determinant of this 2x2 sub-matrix.

Then, multiply each element by its cofactor. Remember, there's an alternating sign pattern, (+, -, +, ...) when expanding. Sum up the results, and voila! You have the determinant.

Matrix Operations

Understanding matrix operations is crucial in working with matrices and includes operations like addition, subtraction, and multiplication. Here we focus on those as they relate to finding a determinant.

• Matrix multiplication is essential for finding cofactors. Once you've isolated the 2x2 matrix, calculate its determinant by multiplying diagonally and taking the difference: a*d - b*c. This operation helps us get cofactors during the expansion process.
• Keep in mind the order of operations is vital: always remember PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) even when dealing with matrices.
• Finally, remember that while operations can seem tedious, they're necessary to ensure accuracy in your calculations. Accuracy builds confidence and precision in your approach to these mathematical challenges.

By understanding these operations, we can effectively navigate any matrix-related problem.

Graphing Utility Verification

Graphing utilities can be fantastic tools for verifying the calculations you performed manually. This is handy, particularly when dealing with more complex matrices.

• A graphing utility can graphically represent matrix calculations, making it easier to catch potential errors.
• These tools offer a quick way to double-check results. Once you've completed your determinant calculation, simply input the matrix into your graphing utility and compare the outputs.
• They can also help students visualize the matrices, offering another way to understand their structure and function beyond just numbers on a page.

In our example, using a graphing utility effectively confirmed that the determinant of our 3x3 matrix was indeed zero. These tools are not just cheats but essential companions for verifying your work and ensuring it's accurate.