Problem 9 - Algebra 2

Question

Find the value of each determinant. $$ \left|\begin{array}{cc}{10} & {6} \\ {5} & {5}\end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is 20.

Step 1: Understand the Determinant Formula for a 2x2 Matrix

The determinant of a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is calculated using the formula \( ad - bc \). This means you multiply the elements on the main diagonal (top-left to bottom-right) and then subtract the product of the elements on the other diagonal (top-right to bottom-left).

Step 2: Identify the Matrix Elements

From the matrix \( \begin{pmatrix} 10 & 6 \ 5 & 5 \end{pmatrix} \), identify each element: \( a = 10 \), \( b = 6 \), \( c = 5 \), and \( d = 5 \).

Step 3: Apply the Determinant Formula

Substitute the identified elements into the formula: \[ ad - bc = 10 \times 5 - 6 \times 5 \].

Step 4: Perform the Calculations

Calculate the products: \( 10 \times 5 = 50 \) and \( 6 \times 5 = 30 \). Subtract the second product from the first: \( 50 - 30 = 20 \).

Step 5: State the Result

The determinant of the given matrix is \( 20 \).

Key Concepts

2x2 matrix

A 2x2 matrix is a simple yet powerful mathematical structure used in various areas such as algebra, geometry, and computer science. Essentially, it is a square array comprising two rows and two columns, thus containing four elements overall. For instance, the matrix \[\begin{pmatrix} a & b \ c & d \end{pmatrix}\]shows a typical 2x2 arrangement. Here, each position in the matrix has specific coordinates that we refer to using row and column indices. This setup forms the basis for many operations, including the important task of finding determinants. Understanding how to navigate and manipulate these small yet significant matrices is crucial for solving related problems efficiently.

matrix elements

In the context of a 2x2 matrix, the elements are simply the numbers that occupy each of the four positions. For a matrix like \( \begin{pmatrix} 10 & 6 \ 5 & 5 \end{pmatrix} \), the individual elements are denoted as:

• \( a = 10 \), the element in the first row and first column,
• \( b = 6 \), the element in the first row and second column,
• \( c = 5 \), the element in the second row and first column,
• \( d = 5 \), the element in the second row and second column.

These elements can represent different data, depending on the matrix's purpose. In mathematics, precisely identifying and understanding the role of each matrix element is crucial for performing calculations such as finding a determinant.

determinant formula

The determinant of a 2x2 matrix, denoted with vertical bars like \( \left|\begin{array}{cc} a & b \ c & d \end{array}\right| \), is a special value that can tell us important properties about the matrix. The formula to find the determinant of a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is:\[det = ad - bc\]This formula

• involves multiplying the diagonal elements, \( a \) and \( d, \)
• and subtracting the product of the off-diagonal elements, \( b \) and \( c. \)

Understanding this order—main diagonal first, then the other diagonal—is key to calculating the determinant correctly.

calculations

To find the determinant of the matrix \( \begin{pmatrix} 10 & 6 \ 5 & 5 \end{pmatrix} \), begin by applying the determinant formula. Start with identifying each element, then calculate the following:

• Compute the product of the main diagonal elements: \( 10 \times 5 = 50. \)
• Calculate the product of the other diagonal elements: \( 6 \times 5 = 30. \)
• Subtract the second product from the first to get the determinant: \( 50 - 30 = 20. \)

These steps demonstrate both the subtractive nature of determinant calculations and how precision in arithmetic can yield insightful results. The determinant here, \( 20, \)conveys that this matrix is non-degenerate, making it invertible in mathematical contexts.