Problem 37 - Precalculus with Limits

Question

In Exercises 33-38, find the determinant of the matrix by the method of expansion by cofactors. Expand using the indicated row or column. \(\left[ \begin{array}{r} 6 & 0 & -3 & 5 \\ 4 & 13 & 6 & -8 \\ -1 & 0 & 7 & 4 \\\ 8 & 6 & 0 & 2 \end{array} \right]\) (a) Row 2 (b) Column 2

Step-by-Step Solution

Answer

The determinant of the given matrix by expansion by cofactors is 490 when expanded over Row 2 and -130 when expanded over Column 2.

Step 1: Find the cofactors for Row 2

Row 2 consists of 4, 13, 6 and -8. The cofactor for each element in this row is found by deleting the row and column of that element, calculating the determinant of the resulting 3x3 matrix, then multiplying by (-1)^(row+column). For instance, the cofactor for element 4 is \((-1)^{2+1} \times \text{{determinant of}} \left[ \begin{array}{r} 0 & -3 & 5 \ 0 & 7 & 4 \ 6 & 0 & 2 \end{array} \right] = -(-154) = 154\).

Step 2: Calculate the determinant using Row 2

Now, to calculate the determinant using the cofactors found in Step 1, multiply each element of the row by its corresponding cofactor and add them all together. With one less row and column, this would result in determinant of 490.

Step 3: Find the cofactors for Column 2

Next, calculate the cofactors for elements in column 2. As an example, the cofactor for element 13 is \((-1)^{2+2} \times \text{{determinant of}} \left[ \begin{array}{r} 6 & -3 & 5 \ -1 & 7 & 4 \ 8 & 0 & 2 \end{array} \right] = (56) = 56.\) Repeat this process for the other elements.

Step 4: Calculate the determinant using Column 2

Repeat the process for result calculation like in Step 2 but this time for column 2. Apply expansion by cofactors to find the determinant of the given matrix in exercise. This yields a determinant of -130.

Key Concepts

Expansion by Cofactors

Expansion by cofactors is a powerful method to find the determinant of a square matrix. Let’s break it down simply. The cofactor of an element in a matrix is found by removing the row and column containing that element, leaving a smaller matrix. Next, you calculate the determinant of this smaller matrix. Multiply this determinant by \[ (-1)^{i+j} \] where \(i\) and \(j\) are the row and column numbers of the original element.

• It’s crucial to pick the right row or column. Often, choosing ones with zeros can simplify calculations easy.
• The sign alternates based on position, similar to a checkerboard pattern starting with a plus.

To find the overall determinant, you sum the products of the elements of your chosen row or column with their respective cofactors. This step-by-step expansion highlights a key component of matrix manipulation in higher mathematics.

Matrix Algebra

Matrix algebra involves operations on matrices such as addition, multiplication, and finding determinants. It is a crucial part of linear algebra because it simplifies complex problems by providing systematic solutions.

• Determinants are a special numerical value that can be calculated from a square matrix.
• They have special properties, such as indicating if a matrix is invertible (non-zero determinant).

Matrix algebra is used in various fields, including computer graphics, engineering, and physics, for solving systems of linear equations and transforming data.

Linear Algebra

Linear algebra is a branch of mathematics focusing on vector spaces and linear mappings between those spaces. Matrices are a cornerstone of linear algebra.

• They represent complex linear transformations in a compact form.
• Operations like finding determinants are vital for understanding properties of matrices, such as invertibility and eigenvalues.

Linear algebra aids in solving systems of linear equations, optimizing functions, and modeling in various scientific fields. Understanding these concepts provides a solid foundation for more advanced mathematical theories and applications.