Problem 7 - Intermediate Algebra

Question

Fill in the blanks. In evaluating the determinant below, about what row or column was it expanded? $$ \left|\begin{array}{rrr} 5 & 1 & -1 \\ 8 & 7 & 4 \\ 9 & 7 & 6 \end{array}\right|=-1\left|\begin{array}{ll} 8 & 7 \\ 9 & 7 \end{array}\right|-4\left|\begin{array}{ll} 5 & 1 \\ 9 & 7 \end{array}\right|+6\left|\begin{array}{ll} 5 & 1 \\ 8 & 7 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is expanded about the third column.

Step 1: Identify the Original Matrix Format

The original matrix is a 3x3 matrix, given as \(\begin{bmatrix} 5 & 1 & -1 \ 8 & 7 & 4 \ 9 & 7 & 6 \end{bmatrix}\). The task is to determine about which row or column this determinant is being expanded.

Step 2: Analyze the Expanded Determinant Expression

The expanded form of the determinant is given as: \(-1\begin{vmatrix} 8 & 7 \ 9 & 7 \end{vmatrix} - 4\begin{vmatrix} 5 & 1 \ 9 & 7 \end{vmatrix} + 6\begin{vmatrix} 5 & 1 \ 8 & 7 \end{vmatrix}\).Each minor in this expression corresponds to removing one row and one column from the original matrix.

Step 3: Match the Terms to the Original Matrix Elements

- The first term \(-1\begin{vmatrix} 8 & 7 \ 9 & 7 \end{vmatrix}\) indicates removing the first row and third column.- The second term \(-4\begin{vmatrix} 5 & 1 \ 9 & 7 \end{vmatrix}\) indicates removing the second row and third column.- The third term \(+6\begin{vmatrix} 5 & 1 \ 8 & 7 \end{vmatrix}\) indicates removing the third row and third column.This pattern shows that the expansion involves different rows but the same column.

Step 4: Determine the Column About Which the Expansion is Done

Since all the terms correspond to removing the third column (the third column is constant in each minor term), the determinant is expanded about the third column of the matrix (values: -1, 4, and 6).

Key Concepts

Matrix Expansion

Matrix expansion, also known as cofactor expansion, is a key concept in the calculation of determinants for matrices. This method involves expanding a determinant along a specific row or column. When you expand along a row or column, you express the determinant of the matrix in terms of its smaller 2x2 determinants, known as minors.

The purpose of matrix expansion is to simplify the calculation of a determinant, especially for matrices larger than 2x2, such as a 3x3 matrix. To expand the determinant of a 3x3 matrix like the one we're examining, you choose one row or column to perform the expansion. Each element of the chosen row or column is multiplied by its cofactor, which is positive or negative depending on its position, and is determined by a smaller 2x2 determinant after removing the corresponding row and column.

Using expansion along a column or row allows mathematicians to calculate determinants more easily, breaking down a potentially complicated task into manageable steps, while maintaining accuracy by carefully choosing the row or column that simplifies the arithmetic.

Minors

Minors play a pivotal role in determinant calculations. A minor of an element in a matrix is the determinant of the smaller matrix that remains after removing the row and column containing that element. In our example of the 3x3 matrix, each of the components -1, 4, and 6 in the third column leads to the calculation of a minor.

For instance:

• -1's minor: Remove the first row and third column, leaving the submatrix: \[\begin{bmatrix}8 & 7 \9 & 7\end{bmatrix}\]
• 4's minor: This corresponds to removing the second row and third column, yielding: \[\begin{bmatrix}5 & 1 \9 & 7\end{bmatrix}\]
• 6's minor: Removing the third row and third column gives: \[\begin{bmatrix}5 & 1 \8 & 7\end{bmatrix}\]

Minors are essential because they simplify the computation process of larger determinants by breaking them down into mere 2x2 matrices. With 3x3 or larger matrices, identifying and computing minors allows you to apply simpler arithmetic to determine the original determinant.

3x3 Matrix

A 3x3 matrix is a square array composed of three rows and three columns, making it a standard model in linear algebra for performing various operations, including determinant calculation through expansion. In our example, the 3x3 matrix is:\[\begin{bmatrix}5 & 1 & -1 \8 & 7 & 4 \9 & 7 & 6\end{bmatrix}\]

When working with a 3x3 matrix, remember that it:

• Contains 9 elements altogether.
• Can be reduced into smaller matrices to calculate determinants through minors and cofactor expansion.
• Represents a useful size for many basic matrix operations, situated between simple (2x2 matrices) and complex (larger matrices).

Analyzing a 3x3 matrix through cofactor expansion is manageable and often serves as a stepping stone to understanding more complex matrix manipulations. By mastering determinants of smaller matrices, you build a foundation for handling matrices of any size in your mathematical journey.