Problem 13 - Precalculus: Mathematics for Calculus

Question

Find the determinant of the matrix, if it exists. $$\left[\begin{array}{ll} \frac{1}{2} & \frac{1}{8} \\ 1 & \frac{1}{2} \end{array}\right]$$

Step-by-Step Solution

Answer

The determinant is \( \frac{1}{8} \).

Step 1: Identify the Matrix

The given matrix is a 2x2 matrix: \[\begin{bmatrix}\frac{1}{2} & \frac{1}{8} \1 & \frac{1}{2}\end{bmatrix}\]

Step 2: Recall the Determinant Formula for 2x2 Matrices

For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated as \( ad - bc \).

Step 3: Assign Values to Variables

Assign the elements of the matrix to the variables in the determinant formula: \( a = \frac{1}{2} \), \( b = \frac{1}{8} \), \( c = 1 \), and \( d = \frac{1}{2} \).

Step 4: Plug Values into the Formula

Substitute the values into the determinant formula:\[ \frac{1}{2} \times \frac{1}{2} - \frac{1}{8} \times 1 \]

Step 5: Calculate Each Part

Calculate each part of the expression:- \( \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \)- \( \frac{1}{8} \times 1 = \frac{1}{8} \)

Step 6: Subtract the Products

Subtract the products from Step 5: \[ \frac{1}{4} - \frac{1}{8} \]

Step 7: Find a Common Denominator

Convert \( \frac{1}{4} \) to \( \frac{2}{8} \) to have a common denominator with \( \frac{1}{8} \).

Step 8: Perform the Subtraction

Subtract the fractions: \[ \frac{2}{8} - \frac{1}{8} = \frac{1}{8} \]

Step 9: State the Determinant

The determinant of the matrix is \( \frac{1}{8} \).

Key Concepts

2x2 matrix

A 2x2 matrix is the simplest form of a matrix that still allows us to perform various matrix operations including finding the determinant. A matrix is essentially a rectangular array of numbers arranged in rows and columns. In a 2x2 matrix, we have exactly two rows and two columns. For example, the matrix given in the original exercise is: \[\begin{bmatrix}\frac{1}{2} & \frac{1}{8} \1 & \frac{1}{2}\end{bmatrix}\]Breaking this down:

• The first element in the first row is \( \frac{1}{2} \)
• The second element in the first row is \( \frac{1}{8} \)
• The first element in the second row is \( 1 \)
• The second element in the second row is \( \frac{1}{2} \)

Understanding this basic structure is crucial for performing operations like finding determinants. 2x2 matrices are often used to solve systems of linear equations and represent transformations in two-dimensional space.

matrix determinants

The determinant is a special number that can be calculated from a square matrix. For a 2x2 matrix, such as \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant provides important insights into the matrix's properties.The formula for finding the determinant of a 2x2 matrix is:\[ \text{det}(A) = ad - bc\]This simple formula allows you to quickly calculate the determinant using the elements of the matrix arranged as: top-left \(a\), top-right \(b\), bottom-left \(c\), and bottom-right \(d\).

• If the determinant is zero, it means the matrix is singular, and it does not have an inverse.
• If the determinant is not zero, the matrix is invertible, which means it can be used to solve linear equations.

Determining whether a matrix is invertible is crucial in many applications of linear algebra, such as determining linear independence and solving systems of equations.

linear algebra

Linear algebra is a branch of mathematics concerned with vectors, vector spaces, and linear equations. Matrices, like the 2x2 matrix, are fundamental entities in linear algebra, representing linear transformations and systems of linear equations. In linear algebra, determinants play a key role:

• They help in understanding whether a system of linear equations has a unique solution, infinitely many solutions, or no solutions.
• A non-zero determinant indicates that vectors (rows or columns of a matrix) are linearly independent, meaning they do not lie on the same line or plane.
• Determinants are used in calculating eigenvalues, which are important in the study of differential equations and many applications in physics and engineering.

Linear algebra is essential in various fields, including computer science, physics, engineering, and economics, as it provides tools for modeling and solving complex problems in multiple dimensions.