Problem 2 - Precalculus

Question

Compute the determinant of the given matrix. (Some of these matrices appeared in Exercises \(1-8\) in Section 8.4.) \(C=\left[\begin{array}{rr}6 & 15 \\ 14 & 35\end{array}\right]\)

Step-by-Step Solution

Answer

The determinant of matrix \( C \) is 0.

Step 1: Identify the Matrix Type

The given matrix \( C \) is a 2x2 matrix. When computing the determinant for a 2x2 matrix with elements \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), we use the formula \( ad - bc \).

Step 2: Assign Matrix Elements

Assign the elements from matrix \( C \): let \( a = 6 \), \( b = 15 \), \( c = 14 \), and \( d = 35 \).

Step 3: Substitute Elements into Formula

Using the determinant formula for a 2x2 matrix, substitute the values: \( 6 \cdot 35 - 15 \cdot 14 \).

Step 4: Perform Multiplications

Calculate the products: \( 6 \cdot 35 = 210 \) and \( 15 \cdot 14 = 210 \).

Step 5: Compute Determinant

Subtract the two products: \( 210 - 210 = 0 \).

Key Concepts

2x2 Matrix

A 2x2 matrix is a simple and fundamental concept in linear algebra. It refers to a matrix with 2 rows and 2 columns, containing four elements in total. Each of these elements can be a number or a variable. For instance, in our exercise, the matrix \( C \) is identified as a 2x2 matrix, looking like this:

• First row: \([6, 15]\)
• Second row: \([14, 35]\)

These matrices are often used in various calculations and serve as an introduction to more complex matrix forms. By understanding the structure and properties of a 2x2 matrix, students can tackle initial matrix operations with confidence.

Matrix Operations

Matrix operations involve various algebraic processes such as addition, subtraction, multiplication, and finding the determinant or inverse of a matrix. In our exercise, we focused on finding the determinant, a crucial matrix operation that provides insightful information about the matrix.

• To compute the determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), use the formula \( ad - bc \).
• This operation tells us about the linear independence of the matrix's rows and columns.

In our problem, the operation was simple yet powerful, leading us to a determinant of zero, indicating that the rows of the matrix are linearly dependent.

Precalculus Mathematics

Precalculus mathematics bridges the gap between algebra and calculus, introducing students to advanced math concepts, including matrix theory. Understanding matrices, especially operations like determining a determinant, is vital in this field as it applies to solving systems of equations and transformations.

• Such skills are foundational for tackling calculus problems efficiently.
• By mastering matrix operations, students enhance their ability to handle real-life problems in engineering, physics, and economics.

Precalculus equips learners with the necessary tools and confidence to advance into more intricate areas of math.

Algebraic Concepts

Algebraic concepts form the core of understanding matrices and their applications. Working with matrices involves various algebraic manipulations, such as multiplication and subtraction, as seen in our determination of the matrix's determinant.

• Understanding these basics is crucial for further study in mathematics and science.
• The exercise emphasizes algebraic processes like assigning variables, using formulas, and performing arithmetic operations, reinforcing core algebra skills.

Algebra is not just solving equations but understanding how to systematically manipulate symbols and numbers to find solutions, a skill fundamentally linked to working with matrices.