Problem 10 - Intermediate Algebra

Question

Evaluate each of the following determinants. $$ \left|\begin{array}{rr} -2 & 4 \\ 9 & -7 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is -22.

Step 1: Understanding the Determinant Formula

To find the determinant of a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), you can use the formula: \( ad - bc \). This involves multiplying the diagonal elements and then subtracting the product of the other two elements.

Step 2: Identify Elements of the Matrix

The matrix you provided is \( \begin{pmatrix} -2 & 4 \ 9 & -7 \end{pmatrix} \). Here, \( a = -2 \), \( b = 4 \), \( c = 9 \), and \( d = -7 \). Identify these values to use in the determinant formula.

Step 3: Apply the Determinant Formula

Substitute the identified values into the determinant formula: \( ad - bc = (-2)(-7) - (4)(9) \). Compute the products of the diagonal elements and the other two elements.

Step 4: Calculate Each Product

First, calculate the product of the diagonal elements: \((-2) \times (-7) = 14\). Then, calculate the product of the other two elements: \(4 \times 9 = 36\).

Step 5: Subtract the Products

Subtract the second product from the first: \(14 - 36 = -22\). This provides the value of the determinant.

Key Concepts

Matrix Determinant Calculation

Calculating the determinant of a matrix is a fundamental concept in linear algebra. It provides important information about the matrix, such as whether it is invertible and the volume scaling factor represented by the transformation associated with the matrix. Determinants are particularly powerful in solving systems of linear equations and understanding linear transformations.
To calculate the determinant of a 2x2 matrix, you can use the formula:

• Identify the elements of the matrix, typically represented as \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \).
• Apply the determinant formula: \( ad - bc \).
• The result from this calculation gives you the determinant value.

The determinant represents the scaling factor of the transformation described by the matrix; if it is zero, the matrix does not have an inverse and the transformation "collapses" space.

2x2 Matrices

A 2x2 matrix is one of the simplest types of matrices in linear algebra, consisting of two rows and two columns. Despite its simplicity, it plays a crucial role in understanding more complex concepts.
For a matrix \( \begin{pmatrix} -2 & 4 \ 9 & -7 \end{pmatrix} \), it's essential to understand how each element contributes to various operations, like the determination of its determinant. Each element is positioned specifically to form a square, which is foundational for matrix operations.A 2x2 matrix helps to:

• Simplify the visualization of linear transformations in two-dimensional space.
• Provide a basis for learning about larger matrices and their operations in matrix algebra.

By mastering the determinant of a 2x2 matrix, learners can build confidence to explore more advanced topics within the field.

Matrix Algebra

Matrix algebra is a branch of mathematics focusing on matrix operations and their properties. Understanding the properties of matrices, such as determinants, is central to many areas, including physics, engineering, and computer science. Key components of matrix algebra include:

• Matrix addition and subtraction, where matrices of the same size are combined by adding or subtracting corresponding elements.
• Matrix multiplication, which combines two matrices to form another matrix, focusing on dot products of rows and columns.
• The computation of determinants, especially for square matrices, to analyze properties related to linear equations and transformations.

By grasping the basic matrix algebra concepts, students can tackle problems involving various transformations and theoretical applications, setting a strong foundation for more complex studies in linear algebra.