Problem 53 - Precalculus with Limits

Question

In Exercises 39-54, find the determinant of the matrix.Expand by cofactors on the row or column that appears to make the computations easiest. \(\left[ \begin{array}{r} 3 & 2 & 4 & -1 & 5 \\ -2 & 0 & 1 & 3 & 2 \\ 1 & 0 & 0 & 4 & 0 \\ 6 & 0 & 2 & -1 & 0 \\ 3 & 0 & 5 & 1 & 0 \end{array} \right]\)

Step-by-Step Solution

Answer

The determinant of the given matrix is 0.

Step 1: Setting up expansion

We will expand along the second column as it contains three zeros. The elements from the second column will multiply the minors of the matrix reduced by one row and one column. Then, we will alternate the signs which yields: \( 2 \cdot M_{12} - 0 \cdot M_{22} + 0 \cdot M_{32} - 0 \cdot M_{42} + 0 \cdot M_{52} \), where M_{ij} represents the minor of the element at the ith row and jth column.

Step 2: Compute the first minor

We have to calculate the determinant of the 4x4 matrix obtained by removing the first row and the second column. This yields \( M_{12} = \left| \begin{array}{r} -2 & 1 & 3 & 2 \ 1 & 0 & 4 & 0 \ 6 & 2 & -1 & 0 \ 3 & 5 & 1 & 0 \end{array} \right| \). The simplest way to compute this determinant is by applying cofactor expansion again along the third column of this sub-matrix. This leads to: \( \left|-2 * 5 * 0 * 0 + 1 * -1 * 3*0 + 3*2*0*5 - 2*1*2*0\right| = 0 \)

Step 3: Substituting the minor

Substituting the value of the first minor \( M_{12} \) in the initial equation, we get \( 2 \cdot 0 - 0\cdot M_{22} + 0 \cdot M_{32} - 0 \cdot M_{42} + 0 \cdot M_{52} = 0 \)

Key Concepts

Cofactor Expansion

Cofactor expansion is a powerful technique used to compute the determinant of a matrix. Essentially, it involves expanding the matrix along a specified row or column, which simplifies the process. This method is especially beneficial when a row or column contains several zeros.
Here's the basic idea behind cofactor expansion:

• Select a row or column to expand upon. Choosing one with multiple zeros reduces calculations.
• Each element of the row or column is multiplied by its cofactor, which is the minor of the element, adjusted with a sign determined by the position.
• The signs alternate in a checkerboard pattern starting with a positive (+) for the first element.
• Sum these cofactor terms to get the determinant.

In our problem, expanding along the second column exploits the zeros present, reducing significant computation. Remember, each zero directly renders corresponding cofactor calculations unnecessary.

Matrix Minors

A matrix minor is a key player in the computation of determinants when using cofactor expansion. For any given element in a matrix, its minor is represented by the determinant of a smaller matrix. This smaller matrix is formed by removing the row and column where the element is located.

For example, in the original matrix given, if an element is located at the first row and the second column, its minor involves deleting that row and column and finding the determinant of the resulting 4x4 submatrix.
Matrix minors might seem small, but their computation is important:

• To find the minor, remove the specific row and column from the original matrix.
• Calculate the determinant of the smaller, resulting matrix, which becomes the minor.

In our example, the calculation of the minor for the element in the first row and second column involved creating a new 4x4 matrix. However, further inspection of this minor via another row with abundant zeros illustrated the computation's simplicity.

Computation of Determinants

Computing determinants can initially seem daunting, especially for larger matrices. But with structured techniques like cofactor expansion, this task becomes manageable. The determinant is a special number that gives a lot of information about the matrix, including whether it is invertible.

Here are steps to compute determinants effectively:

• Identify a row or column to perform cofactor expansion. Prioritize ones with zeros to streamline calculations.
• Calculate minors for each non-zero element, and multiply by the element and the proper sign.
• Add these values to compile the determinant of the matrix.

Our example demonstrated this process well, utilizing a clever expansion along a column full of zeros. The critical observation was that many terms vanished due to multiplication by zero, displaying the efficiency of choosing strategically situated rows or columns.