Problem 9 - College Algebra with Corequisite Support

Question

For the following exercises, find the determinant. $$ \left|\begin{array}{rr} 1 & 0 \\ 3 & -4 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is -4.

Step 1: Identify the Formula

The determinant of a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is calculated using the formula: \( ad - bc \). This formula requires multiplying the diagonal elements and subtracting the product of the off-diagonal elements.

Step 2: Identify the Matrix Elements

In the given matrix, \( a = 1 \), \( b = 0 \), \( c = 3 \), and \( d = -4 \). We will use these values in the formula to find the determinant.

Step 3: Apply the Formula

Substitute the identified values into the determinant formula: \( 1 \cdot (-4) - 0 \cdot 3 \). First, calculate \( 1 \cdot (-4) \) which equals \(-4\). Then, calculate \( 0 \cdot 3 \) which equals 0. Thus, the calculation becomes \(-4 - 0\).

Step 4: Calculate the Determinant

Compute the final result: \(-4 - 0 = -4\). Therefore, the determinant of the given matrix is \(-4\).

Key Concepts

Matrix Elements

To solve problems involving matrices, recognize their individual components, also known as elements. In a 2x2 matrix structure, which looks like: \[ \begin{pmatrix} a & b \ c & d \end{pmatrix} \]we identify each element by its position within the matrix. The elements are:

• **a**: Upper-left element
• **b**: Upper-right element
• **c**: Lower-left element
• **d**: Lower-right element

Understanding matrix elements allows you to plug these values into formulas to perform calculations such as determinants. In our particular matrix, \(a = 1\), \(b = 0\), \(c = 3\), and \(d = -4\). Knowing these allows us to accurately proceed with calculations.
Remember that correct identification of each element is crucial as misplacing any value can lead to incorrect results.

Formula Application

Once we've identified the matrix elements, the next step involves applying the formula specific to finding the determinant of a 2x2 matrix. The formula is:
\[ ad - bc \]This formula breaks down as follows:

• First, you multiply the elements on the main diagonal (top-left to bottom-right), denoted as \(a\) and \(d\).
• Second, subtract the product of the elements on the other diagonal (top-right to bottom-left), denoted as \(b\) and \(c\).

For our example, substitute \(a = 1\), \(b = 0\), \(c = 3\), \(d = -4\) into the determinant formula:
\[ (1)(-4) - (0)(3) \]This simplifies to:

• \((-4) - 0 = -4\)

By applying the formula accurately, we determine the value of the determinant, which is a single numeric result.

Matrix Operations

Matrix operations extend beyond just identifying elements and applying formulas. They involve a series of mathematical procedures that help us understand and work with matrices. In the context of calculating the determinant, we're performing a series of operations:

• **Multiplication**: Multiply the diagonal elements to find parts of the determinant calculation.
• **Subtraction**: Subtract the products of diagonal elements to compute the final determinant.

These operations, although simple, are foundational in understanding more complex matrix manipulations in higher dimensional matrices or more advanced topics like matrix inverses or eigenvalues.
Practicing these operations helps build intuition and prepares students for tackling larger matrices with more intricate operations. The process of finding the determinant essentially reinforces the understanding of these basic yet powerful matrix operations.