Problem 9 - College Algebra

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 Matrix Size

The given matrix is a 2x2 matrix. A 2x2 matrix has the format \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \]. In this particular matrix, the elements are: \( a = 1, b = 0, c = 3, d = -4 \).

Step 2: Apply the 2x2 Determinant Formula

The determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is calculated using the formula: \( \text{det} = ad - bc \). Substitute the identified values: \( a = 1, b = 0, c = 3, d = -4 \).

Step 3: Compute the Determinant

Substitute the values into the formula: \[ \text{det} = (1)(-4) - (0)(3) \]. Calculate the result: \( \text{det} = -4 - 0 = -4 \).

Key Concepts

Understanding a 2x2 Matrix

A 2x2 matrix is essentially a small table consisting of two rows and two columns. It is represented as \[\begin{bmatrix} a & b \ c & d \end{bmatrix} \]where each letter represents a number or an element of the matrix. In this form:

• is in the first row and first column,
• is the first row and second column,
• is in the second row and first column,
• and sits in the second row and second column.

In our original exercise, the matrix given was:\[\left[\begin{array}{rr}1 & 0 \3 & -4\end{array} \right]\]Here, we identify our elements as:

• = 1
• = 0
• = 3
• = -4

This basic understanding of the 2x2 format is crucial before moving ahead with calculating determinants or performing other matrix operations. Knowing which number corresponds to which position aids clarity in calculations.

Determinant Formula of a 2x2 Matrix

Determinants are special numbers that can be calculated from a square matrix. For a 2x2 matrix like the one we have, the determinant is beautifully simple to compute.The general determinant formula for a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is:\[\text{det} = ad - bc\]This formula instructs us to multiply the elements diagonally and then subtract the products:

• Multiply the element by .
• Multiply the element by .
• Subtract the second product from the first one.

Using the numbers from our matrix:\[\text{det} = (1)(-4) - (0)(3) \]This simplifies to:\[\text{det} = -4 - 0 = -4\]The determinant is a scalar value and not a matrix. It holds significant meaning in linear algebra, such as indicating if matrices are invertible or not.

Matrix Operations Involving Determinants

Matrix operations are procedures that can be performed on matrices, including addition, subtraction, multiplication, and finding determinants. When dealing with 2x2 matrices, understanding how to compute a determinant is foundational. Here's why determinants matter in matrix operations:

• **Inverses:** A matrix can only have an inverse if its determinant is non-zero. Thus, calculating the determinant helps determine invertibility.
• **System of Equations:** Determinants can be used in Cramer's Rule to solve linear systems.
• **Eigenvalues and Eigenvectors:** Determinants are tied closely to these concepts, which are essential for understanding linear transformations.

Performing operations with matrices can initially seem complex, but sticking to essential formulas like the one for determinants simplifies processes. In the matrix world, everything is interconnected. Mastery of basic calculations like determinants leads to greater ease in more advanced concepts.