Problem 11 - Algebra 2

Question

Find the value of each determinant. $$ \left|\begin{array}{ll}{-7} & {3} \\ {-9} & {7}\end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is -22.

Step 1: Understand the Determinant of a 2x2 Matrix

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

Step 2: Identify Matrix Elements

For the matrix given: \( a = -7 \), \( b = 3 \), \( c = -9 \), and \( d = 7 \). These values are essential for plugging into the determinant formula.

Step 3: Plug Values into Determinant Formula

Substitute the values into the formula: \[ (-7)(7) - (3)(-9) \].

Step 4: Calculate Products

First, calculate the products: \((-7) \times 7 = -49\) and \(3 \times (-9) = -27\). Ensure you pay attention to the negative signs while performing these calculations.

Step 5: Compute the Determinant

Now apply the determinant formula: \(-49 - (-27) = -49 + 27 = -22\). Therefore, the determinant is -22.

Key Concepts

2x2 Matrix

A 2x2 matrix is a fundamental concept in linear algebra, representing a simple grid with two rows and two columns. This type of matrix is often represented as follows:\[ \begin{pmatrix} a & b \ c & d \end{pmatrix} \]Every 2x2 matrix is characterized by four elements. Each element is positioned uniquely based on its row and column locations. They are designated as:

• \( a \) at the top-left
• \( b \) at the top-right
• \( c \) at the bottom-left
• \( d \) at the bottom-right

This simple structure makes 2x2 matrices easy to handle. Understanding how the elements fit together is key to performing operations like finding the determinant.

Matrix Calculation

Matrix calculations involve various arithmetic operations, but here we are focused on finding the determinant of a 2x2 matrix. The determinant is calculated using the formula:\[ ext{Determinant} = ad - bc \]This basic operation is crucial in many areas of mathematics, such as solving systems of equations and analyzing linear transformations. The calculation requires careful attention to multiplying and subtracting the products of specific elements in the matrix. In our example, for matrix:\[ \begin{pmatrix} -7 & 3 \ -9 & 7 \end{pmatrix} \]We use:

• Multiply the diagonal elements: \((-7) \times 7 = -49\)
• Multiply the other diagonal: \(3 \times (-9) = -27\)
• Subtract these products to find the determinant

Such calculations help in defining properties and characteristics of matrices.

Negative Numbers

Negative numbers often appear in matrix calculations and can affect the outcome significantly. In our matrix, both \(-7\) and \(-9\) are negative numbers. When dealing with negatives:

• Remember the rule: multiplying two negatives yields a positive
• When a negative is multiplied by a positive, the result is negative

In our calculation:

• \((-7) \times 7 = -49\), a negative result
• \(3 \times (-9) = -27\), because multiplying a positive by a negative remains negative

Properly handling negative numbers ensures accuracy, especially when dealing with determinants.

Algebraic Expressions

Algebraic expressions form the backbone of operations like determining a matrix's determinant. An algebraic expression consists of numbers, variables, and arithmetic operations. In matrix determinants, each calculation step may be viewed as an algebraic expression.For example:

• The expression \((-7) \times 7 - (3 \times (-9))\)is a clear demonstration of how algebra is applied.
• First, simplify each multiplication separately.
• Next, substitute back into the main expression for any calculations.

These expressions require careful evaluation and attention to detail. Mistakes can arise if operations are not applied consistently, leading to incorrect results. Understanding how algebraic principles apply to matrices helps in better solving matrix-related problems.