Problem 9 - Precalculus: Mathematics for Calculus

Question

Find the determinant of the matrix, if it exists. $$\left[\begin{array}{rr} 4 & 5 \\ 0 & -1 \end{array}\right]$$

Step-by-Step Solution

Answer

The determinant of the matrix is -4.

Step 1: Identify the Elements of the Matrix

The given matrix is a 2x2 matrix: \[\begin{bmatrix} 4 & 5 \ 0 & -1 \end{bmatrix}\] The elements are: - \( a = 4 \), which is the element in the first row and first column. - \( b = 5 \), which is the element in the first row and second column. - \( c = 0 \), which is the element in the second row and first column. - \( d = -1 \), which is the element in the second row and second column.

Step 2: Apply the Formula for the Determinant of a 2x2 Matrix

The formula for the determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is:\[det = ad - bc\] We will apply this formula to the given matrix.

Step 3: Substitute the Values into the Formula

Substitute the identified elements \( a = 4 \), \( b = 5 \), \( c = 0 \), and \( d = -1 \) into the determinant formula:\[det = (4)(-1) - (5)(0)\]

Step 4: Calculate the Determinant

Calculate the value using the substituted elements:\[det = -4 - 0 = -4\] Hence, the determinant of the given matrix is \( -4 \).

Key Concepts

2x2 Matrix

A 2x2 matrix is one of the simplest forms of a matrix in linear algebra, consisting of two rows and two columns. It is commonly used in mathematics due to its straightforward structure. The matrix is formatted as follows: \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]where each of its four elements can be any number, real or complex. In our context, the 2x2 matrix is:\[\begin{bmatrix} 4 & 5 \ 0 & -1 \end{bmatrix}\]This particular matrix is an example where you have simple integers filling each position.

• The first row contains the elements 4 and 5.
• The second row contains the elements 0 and -1.

These elements play a critical role in matrix operations, especially in determining characteristics like the determinant.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and linear equations, forming the cornerstone of many scientific fields. Central to linear algebra is the concept of matrices, which allow us to perform complex transformations and operations efficiently. One fundamental operation is calculating the determinant of a matrix. In linear algebra, the determinant is a scalar value that provides insight into the properties of a matrix, such as whether a system of linear equations has a unique solution, no solution, or infinitely many solutions. Understanding the determinant helps in solving systems of linear equations, calculating eigenvalues, and transforming geometric spaces. With our 2x2 example: - The process of finding the determinant helps in understanding these various properties and operations. Ultimately, linear algebra's role is crucial in computational science, engineering, computer graphics, and statistics, among other fields.

Matrix Operations

Matrix operations are the various methods used to compute and manipulate matrices within linear algebra. These include addition, subtraction, multiplication, and finding determinants or inverses of matrices. Specifically, determining a matrix's determinant is a key operation that reveals essential information about the matrix.For a 2x2 matrix, the operation to find its determinant follows a simple formula:- Given a matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is computed as \(ad - bc\).Let's apply this to our example:- Substitute in the elements: \(a = 4\), \(b = 5\), \(c = 0\), and \(d = -1\).- The calculation follows: \(det = (4)(-1) - (5)(0) = -4\).This basic operation helps in matrix analysis, contributing to deeper studies in systems of equations and transformation properties. Understanding this allows students to grasp how matrices can be scaled, rotated or even used to solve complex mathematical problems.