Problem 5 - Precalculus: Functions and Graphs

Question

Exer. \(3-12:\) Find the inverse of the matrix if it exists. $$\left[\begin{array}{ll} 2 & 4 \\ 4 & 8 \end{array}\right]$$

Step-by-Step Solution

Answer

The matrix is not invertible because its determinant is 0.

Step 1: Check if the Matrix is Invertible

For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), it is invertible if and only if the determinant is non-zero. The determinant is given by \( ad - bc \). For our matrix \( \begin{bmatrix} 2 & 4 \ 4 & 8 \end{bmatrix} \), we have \( a = 2 \), \( b = 4 \), \( c = 4 \), \( d = 8 \). Calculate the determinant.

Step 2: Calculate the Determinant

Using \( ad - bc \), compute the determinant as follows: \( 2 \times 8 - 4 \times 4 = 16 - 16 = 0 \).

Step 3: Analyze the Deteminant Result

Since the determinant of the matrix is 0, the matrix is not invertible. A matrix is only invertible if the determinant is non-zero.

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special number that gives us important information about the matrix's properties, especially when discussing square matrices. This number is crucial because it helps us understand if a matrix can be inverted or not. For a 2x2 matrix:

• The formula to calculate the determinant is simple: it’s given by \( ad - bc \), where \( a \), \( b \), \( c \), and \( d \) are the elements of the matrix.

To compute the determinant:

• Multiply the two diagonal elements (\( a \) and \( d \)).
• Subtract the product of the other two elements (\( b \) and \( c \)) from it.

In our example matrix \( \begin{bmatrix} 2 & 4 \ 4 & 8 \end{bmatrix} \), this calculation would be \( 2 \times 8 - 4 \times 4 = 0 \). Because the result is zero, this number indicates that the matrix has some special properties, specifically related to its invertibility.

Invertibility of Matrices

Invertibility of matrices is a key concept in linear algebra because it associates directly with solving systems of linear equations. In simple terms:

• A matrix is considered invertible if there exists another matrix that, when multiplied with the original, yields the identity matrix.
• The identity matrix is like 1 for numbers; it doesn’t change anything when used in multiplication.

Determinant’s Role:

For a 2x2 matrix, invertibility is directly tied to its determinant. Only if the determinant is not zero can the matrix have an inverse. If the determinant is zero:

• The matrix is labeled as singular or non-invertible.
• It means there isn't a unique solution if you were solving a system of equations expressed by this matrix.

2x2 Matrices

2x2 matrices are the most basic example of square matrices that we usually learn about first in matrix algebra. They are small, making them ideal for understanding foundational concepts like determinants and invertibility. Here are a few points to remember:

• A 2x2 matrix looks like \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), which makes calculations straightforward.
• They are often used in basic transformations in geometry, such as rotations and reflections.
• Despite their size, they illustrate crucial ideas, like requiring a non-zero determinant for invertibility.

When working with 2x2 matrices:

• Always ensure to check the determinant before trying to find an inverse.
• Use the formula \( ad - bc \) for the determinant; it's simple but decisive in many analyses.

These matrices serve as the foundation, and with these simple rules, we can build upon more complex matrix operations and analyses. Remember, understanding these basics will pave the way for tackling larger matrices and more intricate problems!