Problem 20 - Intermediate Algebra

Question

Evaluate each determinant. $$ \left|\begin{array}{cc} 10 & 15 \\ 15 & 5 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is -175.

Step 1: Identify the elements

In a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated using the formula \( ad - bc \). Here, the elements of the matrix are \( a = 10 \), \( b = 15 \), \( c = 15 \), and \( d = 5 \).

Step 2: Apply the determinant formula

Using the values identified, substitute them into the determinant formula \( ad - bc \). This becomes \( (10)(5) - (15)(15) \).

Step 3: Calculate the products

Calculate each product inside the determinant formula: \( (10)(5) = 50 \) and \( (15)(15) = 225 \).

Step 4: Solve the determinant value

Subtract the product of \( bc \) from the product of \( ad \): \( 50 - 225 = -175 \). The determinant of the matrix is \( -175 \).

Key Concepts

Understanding a 2x2 Matrix

A 2x2 matrix is simply a table with two rows and two columns. It represents a collection of four numbers or elements, arranged in a specific way. Think of it as a little grid that holds numbers for performing various mathematical operations. You can imagine this grid as:
\[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \]The letters \(a, b, c,\) and \(d\) are placeholders for the numbers that you will encounter when working with specific matrices. By looking at a 2x2 matrix, you can see these elements in their distinct spots:

• a is the upper left element
• b is the upper right element
• c is the lower left element
• d is the lower right element

These positions make it easy to visualize and work with calculations like finding their determinants.

Matrix Elements Demystified

Matrix elements are the building blocks of matrices. When dealing with a 2x2 matrix, these four elements hold significant roles in calculations like finding a determinant. Each element specifically contributes to positions in the matrix grid, and knowing them helps you perform accurate calculations.
If your matrix is represented as:\[\begin{bmatrix}10 & 15 \15 & 5\end{bmatrix}\]In this scenario:

• The \(a\) element is 10
• The \(b\) element is 15
• The \(c\) element is 15
• The \(d\) element is 5

Being able to identify each element's numerical value quickly is essential for any further computations, such as using them in the determinant formula.

Calculating the Determinant Using a Formula

The determinant of a 2x2 matrix is a special value that can be calculated using a simple formula. This formula involves the elements \(a, b, c,\) and \(d\) from the matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \). The formula is expressed as:
\[det = ad - bc\]This tells you to multiply the \(a\) element by the \(d\) element and subtract the product of the \(b\) element and \(c\) element. Let’s apply it to a specific example:

• Substitute the elements from your matrix: \(a = 10\), \(b = 15\), \(c = 15\), \(d = 5\).
• Plug them into the formula: \(10 \times 5 - 15 \times 15\).
• Calculate each part: \(10 \times 5 = 50\) and \(15 \times 15 = 225\).
• Complete the subtraction: \(50 - 225 = -175\).

The calculation results in the determinant being \(-175\). This determinant tells you important properties about the matrix, such as if it has an inverse or not. In this case, the negative result may indicate specific characteristics about the transformation represented by this matrix.