Problem 52 - 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. \(\left[ \begin{array}{r} 1 & 4 & 3 & 2 \\ -5 & 6 & 2 & 1 \\ 0 & 0 & 0 & 0 \\\ 3 & -2 & 1 & 5 \end{array} \right]\)

Step-by-Step Solution

Answer

The determinant of the given matrix is 0.

Step 1: Choosing the Row for Expansion

In this calculation, the third row will be selected for expansion as it contains all zeros, which simplifies the calculations.

Step 2: Calculation of the Determinant

The determinant of a 4x4 matrix can be calculated by the rule of Sarrus for 4x4 matrices, or by expanding along a row or column. If expanding along the third row, the formula is \( Det(A) = a_{31}C_{31} + a_{32}C_{32} + a_{33}C_{33}+a_{34}C_{34}\) where \(a_{ij}\) are the elements of the matrix and \(C_{ij}\) are their corresponding cofactors. But since \( a_{31}=a_{32}=a_{33}=a_{34}=0\), the determinant of this matrix is 0.

Key Concepts

Cofactor Expansion

Cofactor expansion, also known as Laplace’s expansion, is a method used to compute the determinant of a matrix. It involves breaking down a larger matrix into smaller matrices, making it particularly handy as the size of the matrix increases. To apply cofactor expansion, you choose a row or a column from the matrix. Then you calculate the determinant of the smaller matrices obtained by removing the chosen row and column element by element. Here's a simple way to think about the steps:

• Select a row or column to expand on (usually the one with the most zeros for simplicity).
• For each element in that row or column, determine its cofactor, which involves computing the determinant of a smaller matrix (also known as a minor), and then multiplying it by (-1) raised to the sum of the row and column indices.
• Multiply each element by its corresponding cofactor, and sum up these results to get the determinant of the original matrix.

In this exercise, since the selected row is full of zeros, the expansion using those entries will directly result in a determinant of zero, greatly simplifying our work.

4x4 Matrix

Understanding the nature of a 4x4 matrix is integral to grasping the calculation of determinants. Unlike a 2x2 or a 3x3 matrix, a 4x4 matrix allows for more complexity, but it follows the same foundational principles. A 4x4 matrix is composed of four rows and four columns, meaning it has 16 elements. The determinant of such a matrix can be calculated using various methods including the cofactor expansion method mentioned before or other techniques like the Rule of Sarrus, typically for smaller matrices. When tackling a matrix of this size, it's effective to:

• Check for rows or columns with zeros to simplify calculations since these make excellent candidates for cofactor expansion.
• Remember that the determinant results from the summation of expanded terms corresponding to either the rows or columns of the matrix.

Dealing with these larger matrices, understanding and applying cofactor expansion effectively allows you to avoid cumbersome calculations and errors.

Zero Row Determinant

A zero row within a matrix provides valuable insights when calculating the determinant. Specifically, if an entire row or column within a matrix is composed of zeros, the determinant of that matrix is zero. This principle is a direct consequence of cofactor expansion. Here's why this happens:

• When performing cofactor expansion, you compute terms involving each element of the chosen row or column, each multiplied by its cofactor, and then summed up.
• If you choose a row or column filled with zeros in the matrix, then every term in the sum is zero since each involves multiplication by a zero element.

Thus, for any matrix where at least one row or column is entirely composed of zeros, regardless of the size of the matrix, the determinant is instantly zero. This makes computations much easier, as seen in this exercise.