Problem 23 - Advanced Engineering Mathematics

Question

In Problems 15-28, evaluate the determinant of the given matrix by cofactor expansion. $$ \left(\begin{array}{lll} 1 & 1 & 1 \\ x & y & z \\ 2 & 3 & 4 \end{array}\right) $$

Step-by-Step Solution

Answer

The determinant is \(2y - x - z\).

Step 1: Identify the Matrix

The given matrix is a 3x3 matrix: \[\begin{pmatrix}1 & 1 & 1 \x & y & z \2 & 3 & 4 \\end{pmatrix}\]

Step 2: Choose a Row or Column for Expansion

To find the determinant by cofactor expansion, choose an easy row or column. Here, we'll expand along the first row for simplicity.

Step 3: Calculate the Cofactors

Calculate the cofactor for each element of the first row.1. The cofactor of element \(1\) at position \((1,1)\) is computed by the determinant of the matrix obtained by deleting its row and column: \[ \begin{vmatrix} y & z \ 3 & 4 \end{vmatrix} = y \cdot 4 - z \cdot 3 = 4y - 3z \]2. The cofactor of element \(1\) at position \((1,2)\) is computed by: \[ \begin{vmatrix} x & z \ 2 & 4 \end{vmatrix} = x \cdot 4 - z \cdot 2 = 4x - 2z \]3. The cofactor of element \(1\) at position \((1,3)\) is computed by: \[ \begin{vmatrix} x & y \ 2 & 3 \end{vmatrix} = x \cdot 3 - y \cdot 2 = 3x - 2y \]

Step 4: Calculate the Determinant

Use the cofactors to compute the determinant. Expand along the first row:\[det(A) = 1 \cdot (4y - 3z) - 1 \cdot (4x - 2z) + 1 \cdot (3x - 2y)\]Simplify:\[det(A) = 4y - 3z - 4x + 2z + 3x - 2y\]Combine like terms:\[det(A) = (4y - 2y) + (-4x + 3x) + (-3z + 2z) = 2y - x - z\]

Key Concepts

Cofactor Expansion

Cofactor expansion is a technique used for calculating the determinant of a square matrix. It involves breaking down a large matrix into smaller matrices, which are easier to handle. This is often called Laplace's expansion. You perform the expansion by choosing a row or a column of the matrix and then calculating the determinant using cofactors of the elements in that selected row or column.
In simple terms, a cofactor is a signed minor. The minor of an element is the determinant of the smaller matrix that remains after removing the element's row and column. The sign of the cofactor is determined by the position of the element: it is positive for positions where the sum of the row and column indices is even, and negative otherwise.

• When choosing a row or column, select one with as many zeroes as possible to simplify calculations.
• Remember that the cofactor expansion applies to both rows and columns.

By methodically applying cofactor expansion, you can simplify complex determinant problems, especially those involving larger matrices, by focusing on the smaller matrices.

3x3 Matrix

A 3x3 matrix is a square matrix, which means it has the same number of rows and columns; in this case, three. It is denoted as:\[\begin{pmatrix}a & b & c \d & e & f \g & h & i \\end{pmatrix}\]
Here each element, such as \(a, b, c,...\) represents a fixed number or in some cases, a variable. The matrix in the given exercise is a 3x3 matrix:\[\begin{pmatrix}1 & 1 & 1 \x & y & z \2 & 3 & 4 \\end{pmatrix}\]
This format is commonly used in various applications within linear algebra, including transformations, systems of equations, and more. Each row or column can represent a vector, and operations on these matrices can reveal properties about these vectors.

Linear Algebra Concepts

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, linear mappings, and systems of linear equations. Understanding these concepts is essential for dealing with determinants and matrices.
Determinants provide scalar values that can summarize certain properties of matrices. They can identify singular matrices, which are matrices that do not have an inverse. In a visual sense, determinants help us understand properties such as scaling factor and rotation.

• Vectors: Objects that have magnitude and direction. In a matrix, rows or columns can represent vectors.
• Vector Space: A collection of vectors that can be added together and multiplied by scalars.
• Systems of Linear Equations: These can be solved using matrix operations, including the computation of determinants.

Grasping these linear algebra foundations enhances problem-solving abilities in fields like applied mathematics, engineering, physics, and computer science.

Matrix Determinant

The determinant of a matrix is a special number that can be calculated from its elements. For a 3x3 matrix, the determinant is derived through a specific arithmetic process involving the matrix's elements.
In the given exercise, the determinant of the matrix, \(\begin{pmatrix}1 & 1 & 1 \x & y & z \2 & 3 & 4 \\end{pmatrix}\), was calculated using cofactor expansion.
The formula for a 3x3 determinant using cofactor expansion along the first row is:\[det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]Where each letter represents an element of the matrix at specific positions:

• First minor involves deleting the first row and first column of the matrix.
• Second minor involves deleting the first row and second column.
• Third minor involves deleting the first row and third column.

The value of the determinant helps determine the properties of the matrix like invertibility and scalar multiplication effects. Essentially, it gives insight into the transformation properties of the linear map represented by the matrix.