Problem 1 - A Graphical Approach to College Algebra

Question

Find each determinant. Do not use a calculator. $$\operatorname{det}\left[\begin{array}{rr}-5 & 9 \\\4 & -1\end{array}\right]$$

Step-by-Step Solution

Answer

The determinant is \(-31\).

Step 1: Identify the Matrix Elements

Given the matrix \( A = \begin{bmatrix} -5 & 9 \ 4 & -1 \end{bmatrix} \), identify the elements: \( a_{11} = -5 \), \( a_{12} = 9 \), \( a_{21} = 4 \), and \( a_{22} = -1 \). These will be used in the formula for the determinant of a 2x2 matrix.

Step 2: Apply the Determinant Formula

Use the formula for the determinant of a 2x2 matrix, which is \( \text{det}(A) = a_{11}a_{22} - a_{12}a_{21} \). Substitute the identified elements into the formula. It becomes \( \text{det}(A) = (-5)(-1) - (9)(4) \).

Step 3: Perform the Multiplications

Calculate each multiplication: \((-5)(-1) = 5\) and \((9)(4) = 36\).

Step 4: Subtract the Products

Subtract the second product from the first: \(5 - 36 = -31\). This gives the determinant of the matrix.

Key Concepts

Matrix Elements

Matrices are essential in various fields such as engineering, physics, and computer science. Understanding matrix elements is the first step in working with matrices. A matrix is essentially a rectangular array of numbers organized in rows and columns. For a 2x2 matrix, like the one in our problem, it's composed of four elements listed as follows:

• The number in the first row and first column is denoted as \(a_{11}\).
• The number in the first row and second column is \(a_{12}\).
• The number in the second row and first column is \(a_{21}\).
• The number in the second row and second column is \(a_{22}\).

In our exercise, the 2x2 matrix given is \(\begin{bmatrix} -5 & 9 \ 4 & -1 \end{bmatrix}\). Consequently, the elements are \(a_{11} = -5\), \(a_{12} = 9\), \(a_{21} = 4\), and \(a_{22} = -1\). Identifying the matrix elements correctly is crucial for further calculations, such as finding the determinant.

Determinant Formula

To find the determinant of a 2x2 matrix, we use a specific formula. The determinant is a special number that can be calculated from a square matrix. It gives important insights into the properties of the matrix, like whether it's invertible. For a 2x2 matrix, the determinant formula is quite simple: \[\text{det}(A) = a_{11}a_{22} - a_{12}a_{21}\]In this exercise, substitute the identified matrix elements into the determinant formula:

• First, multiply \(a_{11}\) and \(a_{22}\): \((-5) \times (-1) = 5\).
• Next, multiply \(a_{12}\) and \(a_{21}\): \(9 \times 4 = 36\).

Finally, according to the formula, subtract the second product from the first: \(5 - 36\). Thus, the determinant of the matrix is \(-31\). This process highlights the elegance of using the determinant formula for calculations in matrix algebra.

Matrix Operations

Matrix operations are essential tools in mathematics, providing ways to handle matrices in practical applications. When working with a matrix, operations such as addition, subtraction, multiplication, and finding the determinant are fundamental. For a 2x2 matrix, calculating the determinant as shown earlier is a basic yet powerful operation.

• Multiplication in matrices often involves multiplying respective elements and summing the products, such as in the determinant formula: \[a_{11}a_{22}\] and \[a_{12}a_{21}\].
• In subtraction, like seen when finding determinants, we subtract one product from another—\(5 - 36\) in our example.

These operations, especially finding the determinant, not only assist in solving mathematical problems but also help in understanding the behavior of more complex systems represented through matrices. It is crucial to comprehend these basic operations to navigate the broader context of matrix algebra.