Problem 3 - Precalculus: Functions and Graphs

Question

Exer. \(3-12:\) Find the inverse of the matrix if it exists. $$\left[\begin{array}{rr} 2 & -4 \\ 1 & 3 \end{array}\right]$$

Step-by-Step Solution

Answer

The inverse matrix is \( \left[\begin{array}{cc} 0.3 & 0.4 \\ -0.1 & 0.2 \end{array}\right] \).

Step 1: Calculate the Determinant

First, ensure the matrix is invertible by calculating its determinant. The determinant of a 2x2 matrix \( \left[\begin{array}{cc} a & b \ c & d \end{array}\right] \) is calculated as \( ad - bc \). For our matrix: \[ 2 \cdot 3 - (-4) \cdot 1 = 6 + 4 = 10. \] Since the determinant is 10, which is non-zero, the matrix is invertible.

Step 2: Find the Adjugate of the Matrix

The adjugate (or adjoint) of our matrix is found by swapping the elements on the main diagonal and changing the signs of the off-diagonal elements. For the given matrix: \[ \text{adjugate} = \left[\begin{array}{cc} 3 & 4 \ -1 & 2 \end{array}\right]. \]

Step 3: Calculate the Inverse Matrix

To find the inverse of the matrix, divide the adjugate matrix by the determinant calculated in Step 1. The formula for the inverse is \( A^{-1} = \frac{1}{\text{det}(A)} \text{adjugate}(A) \). So, the inverse is: \[ \left(\frac{1}{10}\right) \cdot \left[\begin{array}{cc} 3 & 4 \ -1 & 2 \end{array}\right] = \left[\begin{array}{cc} 0.3 & 0.4 \ -0.1 & 0.2 \end{array}\right]. \]

Key Concepts

Determinant Calculation

The determinant is a special number that can help us determine if a matrix is invertible. Not every matrix can have an inverse, but if the determinant is not zero, we know it's possible to find one.
The formula for the determinant of a 2x2 matrix is straightforward. For a matrix like:

• \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \)

The determinant is calculated using:

• \( ad - bc \)

This formula helps us by combining the diagonal products, subtracting one from the other. In our example with the matrix \( \begin{bmatrix} 2 & -4 \ 1 & 3 \end{bmatrix} \), the determinant is:

• \( 2 \times 3 - (-4) \times 1 = 6 + 4 = 10 \)

Since the determinant here is 10, the matrix is invertible. This step is crucial before moving on to calculate the inverse.

Adjugate Matrix

Once you confirm a matrix is invertible, the next step involves finding its adjugate. The adjugate is simply a rearranged form of the original matrix, needed to compute the inverse.
For a 2x2 matrix, the adjugate is determined by:

• Swapping the elements of the main diagonal.
• Changing the signs of the off-diagonal elements.

In our example:

• The original matrix is \( \begin{bmatrix} 2 & -4 \ 1 & 3 \end{bmatrix} \)
• The adjugate becomes \( \begin{bmatrix} 3 & 4 \ -1 & 2 \end{bmatrix} \)

By swapping and changing signs, we've set the matrix up for further operations. This approach is specific to 2x2 matrices but follows a similar guideline for larger ones as well.

2x2 Matrix

The 2x2 matrix is one of the simplest forms of matrices and is widely used in linear algebra. Its structure lays out a grid with two rows and two columns, which makes it perfect for demonstrating basic matrix operations like finding inverses.
A typical 2x2 matrix looks like:

• \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \)

The beauty of the 2x2 matrix lies in its simplicity, allowing straightforward applications of operations like determinant calculation and adjugate formation. All calculations, such as finding an inverse, become more manageable due to its compact size.
Understanding the 2x2 matrix deeply is beneficial, as it's a fundamental building block for exploring more complex matrices.

Inverse of a Matrix

The inverse of a matrix is a crucial concept in solving linear equations and various applications in mathematics and engineering. An inverse matrix, when multiplied by the original matrix, results in the identity matrix.
For a 2x2 matrix, once you have the adjugate and the determinant, you can find the inverse. The formula to calculate the inverse is:

• \( A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adjugate}(A) \)

In practice, this means dividing each element of the adjugate by the determinant. For the matrix \( \begin{bmatrix} 2 & -4 \ 1 & 3 \end{bmatrix} \), after calculating the adjugate and determinant, we find:

• \( A^{-1} = \left(\frac{1}{10}\right) \cdot \begin{bmatrix} 3 & 4 \ -1 & 2 \end{bmatrix} = \begin{bmatrix} 0.3 & 0.4 \ -0.1 & 0.2 \end{bmatrix} \)

This result confirms the inverse, which is useful for matrix operations and provides a deeper understanding of solving matrix equations.