Problem 18 - Algebra for College Students

Question

Evaluate each \(3 \times 3\) determinant. Use the properties of determinants to your advantage. \(\left|\begin{array}{rrr}2 & 35 & 5 \\ 1 & -5 & 1 \\ -4 & 15 & 2\end{array}\right|\)

Step-by-Step Solution

Answer

The determinant is \(-285\).

Step 1: Identify the Matrix

The matrix we need to find the determinant for is a \(3 \times 3\) matrix given by \(\begin{bmatrix} 2 & 35 & 5 \ 1 & -5 & 1 \ -4 & 15 & 2 \end{bmatrix}\). To solve for the determinant, we will use the cofactor expansion along the first row.

Step 2: Calculate First Cofactor

To find the cofactor starting from the first row, first element, we need the sub-matrix formed by deleting the first row and first column, which is \(\begin{bmatrix} -5 & 1 \ 15 & 2 \end{bmatrix}\). The determinant of this sub-matrix is given by \((-5)(2) - (15)(1) = -10 - 15 = -25\), but we will calculate it along with the sign adjusted.

Step 3: Calculate Second and Third Cofactors

For the second element, 35, the sign is negative ( eward{–}1 eward{,3}\). Remove row 1 and column 2 to form the sub-matrix \(\begin{bmatrix} 1 & 1 \ -4 & 2 \end{bmatrix}\). Its determinant is \((1 \times 2) - (1 \times -4) = 2 + 4 = 6\). For the third element, 5, the sub-matrix is \(\begin{bmatrix} 1 & -5 \ -4 & 15 \end{bmatrix}\) with the determinant \((1)(15) - (-5)(-4) = 15 - 20 = -5\, added to the original with positive sign.

Step 4: Apply the Cofactor Expansion

The determinant is given as: \(2 \cdot (-25) + 35 \cdot (-6) + 5 \cdot (-5)\). Simplifying, we get: \((-50) - 210 - 25 = -285\).

Step 5: Simplify the Expression (Verify)

Verify the correctness by recalculation using careful signs, but the following calculation simplifies to a final determinant value:

Key Concepts

3x3 Matrix

A 3x3 matrix is a vital concept in linear algebra, which consists of three rows and three columns. In mathematical notation, such a matrix can be represented as \[ \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \]. In this context, the matrix is fully determined by its nine elements denoted as \( (a, b, c, \ldots, i) \). Each element belongs to a specific position: the first row can be read across the matrix as the sequence \(a, b, c\), while the second and third row are \(d, e, f\) and \(g, h, i\), respectively.
A 3x3 matrix is useful across various fields, including computer science, physics, and engineering, and serves foundational purposes in transformations. Transformations are operations that map coordinates from one space into another using matrix multiplication, a primary tool in matrix algebra.
The structure's simplicity in combination with its computational power makes it an essential building block for more complex mathematical concepts.

Cofactor Expansion

Cofactor expansion, also known as Laplace expansion, is a method used to calculate the determinant of a 3x3 matrix. This method involves selecting any row or column of a matrix and expanding it using the cofactors of its elements. Let's break it down further with an example:
- Step 1: Select a row or column. Typically, the first row is used for simplicity.
- Step 2: Compute the cofactor for each element in the selected row or column. Each cofactor is determined by multiplying the element by \((-1)^{i+j}\), where \(i\) and \(j\) are the row and column indices, respectively. Then, determine the 2x2 submatrix by removing the current row and column of the element, and find its determinant.
- Step 3: Multiply each element in the selected row or column by its cofactor and sum these values to obtain the complete determinant.
For instance, for the matrix given, cofactor expansion along the first row is chosen. Each term in the row provides a partial product that contributes to the overall determinant. This method is powerful because it systematically breaks down complex matrices into smaller, more manageable components.

Determinant Calculation

Calculating the determinant of a 3x3 matrix is a process involving the structured use of the matrix's elements along with their cofactors. The determinant is a special number that can provide numerous insights into a matrix's properties, like whether it is invertible or the volume transformation size it represents.
Here's the method in detail:

• Choose a row or column for expansion. Conveniently, the first row is often selected.
• Calculate the determinant of the 2x2 submatrix for each element in the chosen row or column. You achieve this by crossing out the row and the column of the specific element.
• Apply the sign factor, \((-1)^{i+j}\), to determine each minor's cofactor.
• Sum these terms to find the determinant of the original 3x3 matrix.

This calculation provided in the exercise results in a determinant of -285. The algorithms are directly beneficial in understanding more profound concepts in matrix algebra and emphasizing how inter-related these mathematical principles are. Whether solving a system of linear equations or transforming geometric data, the determinant's role is undeniable.

Matrix Algebra

Matrix algebra forms the foundation for solving a wide variety of mathematical problems involving matrices, including operations such as matrix multiplication, finding inverses, and determinant calculations. Each operation complies with specific rules and procedures intrinsic to the structure and types of matrices involved.
Key components that are crucial within matrix algebra include:

• Matrix Multiplication: Although multiplication of numbers is straightforward, matrix multiplication follows distinct rules that depend on the orientation (rows and columns) of the elements.
• Matrix Inversion: This process involves finding another matrix that, when multiplied by the original matrix, yields the identity matrix.
• Determinants: We've seen how determinant calculations are central to assessing matrix solvability and properties like invertibility.
• Properties and Operations: Such as transposing a matrix, where rows are switched with columns, impacting the subsequent calculations.

Matrix algebra's applications stretch from solving linear systems to applications in physics and computer science, making understanding its concepts, such as determinants and cofactor expansions, vital for students across multiple disciplines.