Problem 7 - College Algebra with Corequisite Support

Question

For the following exercises, find the determinant. $$ \left|\begin{array}{rr} 2 & -5 \\ -1 & 6 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is 7.

Step 1: Identify the elements of the 2x2 matrix

The matrix given is \( \begin{pmatrix} 2 & -5 \ -1 & 6 \end{pmatrix} \). Let the elements be: \( a = 2 \), \( b = -5 \), \( c = -1 \), \( d = 6 \). These correspond to their positions in the matrix.

Step 2: Use the determinant formula for a 2x2 matrix

The formula for the determinant of a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is given by \( ad - bc \).

Step 3: Substitute the values into the formula

By substituting the matrix elements into the determinant formula, we have \( 2(6) - (-5)(-1) \).

Step 4: Calculate each part of the expression

First calculate \( 2 \times 6 = 12 \) and \( -5 \times -1 = 5 \).

Step 5: Subtract the results

Subtract the second result from the first: \( 12 - 5 = 7 \). This is the determinant of the matrix.

Key Concepts

Understanding a 2x2 Matrix

A 2x2 matrix is the simplest type of matrix in terms of square matrices, which means its rows and columns are equal in number. In practical terms, it has two rows and two columns. This structure makes it a foundational building block for more complex matrix operations and helps in understanding larger and more complex matrices. In our example, \[ \begin{pmatrix} 2 & -5 \ -1 & 6 \end{pmatrix} \]we can see that this matrix consists of four elements arranged in two rows and two columns. The simplicity of the 2x2 matrix allows for straightforward calculations, particularly when finding the determinant, a key operation that provides insights into the properties of the matrix.

Defining Matrix Elements

Matrix elements are the individual values in a matrix, indexed by their position within the matrix. Think of a matrix as a grid similar to a spreadsheet, where each spot holds a number or symbol. When we talk about matrix elements in a 2x2 matrix like \[ \begin{pmatrix} 2 & -5 \ -1 & 6 \end{pmatrix} \] we refer to each numerical value inside it:

• \(a = 2\) is the first element at position (1,1)
• \(b = -5\) is the second element at position (1,2)
• \(c = -1\) is the third element at position (2,1)
• \(d = 6\) is the fourth element at position (2,2)

Recognizing and organizing these elements is crucial when applying mathematical operations, such as finding the determinant, because their positions affect the final outcomes.

Using the Determinant Formula

The determinant of a matrix provides valuable mathematical information about the matrix, such as whether it can be inverted or used in solving system equations. For a 2x2 matrix, the determinant is calculated using a specific formula:\[det(A) = ad - bc\]where:

• \( a, b, c, \text{ and } d \) are the matrix elements from positions (1,1), (1,2), (2,1), and (2,2) respectively.

Let's calculate the determinant for our sample matrix:\[ \begin{pmatrix} 2 & -5 \ -1 & 6 \end{pmatrix} \]By inserting the values into the formula:

• First, multiply the main diagonal: \( ad = 2 \cdot 6 = 12 \)
• Then, multiply the other diagonal: \( bc = (-5) \cdot (-1) = 5 \)
• Finally, subtract the results: \( 12 - 5 = 7 \)

So, the determinant of this 2x2 matrix is 7. This straightforward calculation demonstrates the influence of each element's position in the matrix on the determinant's value.