Problem 47 - 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. $$ \left[\begin{array}{rrrr} 3 & 6 & -5 & 4 \\ -2 & 0 & 6 & 0 \\ 1 & 1 & 2 & 2 \\ 0 & 3 & -1 & -1 \end{array}\right] $$

Step-by-Step Solution

Answer

The determinant of the given 4x4 matrix can be calculated using cofactor expansion. Apply the determinant calculation procedure for 3x3 matrices to find D1, and D3, then substitute these values back into the cofactor expansion equation (Step 4) to get the final determinant.

Step 1: Identify the Row or Column for Cofactor Expansion

Avoid rows or columns with numerous non-zero entries since it will lead to complex calculations. Here, the second row of the given matrix is selected as it contains two zero elements, thus simplifying the calculations.

Step 2: Apply the Cofactor Expansion

Cofactor expansion can be listed as follows:\n`-2*(-1)^(1+1)*D1 - 0 + 6*(-1)^(1+3)*D3`, where D1, D3 are 3x3 sub-matrices derived from the original 4x4 matrix by removing the second row and first or third column respectively.

Step 3: Calculate the 3x3 Determinants

Calculate the determinant of the 3x3 matrices, D1=\[\left[\begin{array}{rrr}6 & -5 & 4 \ 1 & 2 & 2 \ 3 & -1 & -1\end{array}\right]\] and D3=\[\left[\begin{array}{rrr}3 & -5 & 4 \ 1 & 2 & 2 \ 0 & -1 & -1\end{array}\right]\] by using formula `a(ei−fh)−b(di−fg)+c(dh−eg)`.

Step 4: Compute the Final Result

Substitute D1 & D3, obtained in the previous step, into the cofactor expansion. Determinant of the 4x4 matrix is `-2*D1 - 6*D3`.

Key Concepts

Cofactor Expansion

Cofactor expansion is a technique used to find the determinant of a square matrix. It is especially useful when dealing with larger matrices, like 4x4 ones. The goal is to select the row or column with the largest number of zeros for expansion. This minimizes the number of calculations, making the evaluation simpler.

When performing cofactor expansion, each element of the chosen row or column is multiplied by its corresponding cofactor. The cofactor is calculated as the signed determinant of the sub-matrix formed by deleting the row and column of the element in question. The sign is determined using a checkerboard pattern, starting with a positive sign in the top-left corner. The formula for cofactor expansion along a row is:

• Det(M) = a(−1)^(i+j)Det(M_ij), where a is the element of the matrix, (i, j) are its row and column indices, and M_ij is the sub-matrix.

Matrix Operations

Matrix operations involve several fundamental processes, including addition, subtraction, multiplication, and finding the determinant. The determinant is a special value that can be calculated from a square matrix. It's pivotal for matrix operations because it helps determine if the matrix is invertible and is used in various applications like solving systems of linear equations.

When executing matrix operations, always follow these key steps:

• Identify the type of operation required for solving your problem.
• Use properties such as zero rows or columns to simplify the computation.
• Always maintain the order of elements as you perform the operations to avoid computational errors.

3x3 Determinant

Calculating a 3x3 determinant is crucial when you're dealing with cofactor expansions of larger matrices. The formula for finding the determinant of a 3x3 matrix helps reduce the complexity of evaluating a 4x4 determinant by breaking it down into smaller parts.

To calculate the determinant of a 3x3 matrix, use the formula:

• Det(A) = a(ei−fh) − b(di−fg) + c(dh−eg)

Where:

• a, b, c are the elements of the first row.
• d, e, f are the elements of the second row.
• g, h, i are the elements of the third row.

Splitting a 4x4 matrix into simpler 3x3 determinants, and evaluating them individually, streamlines the entire calculation process.

Graphing Utility

A graphing utility is a powerful tool that can be used for visualizing matrices and confirming determinant calculations. These applications or calculators simplify the verification process by allowing you to input matrices and quickly get results.

Graphing utilities can perform various tasks:

• They provide visual representations of matrices, making it easier to spot patterns or errors.
• They can quickly compute determinants, reducing human error in arithmetic operations.
• They often offer step-by-step solutions, aiding in your understanding of the procedure.

Using a graphing utility complements manual calculations by confirming your results and providing insights that may not be immediately obvious through traditional computation.