Problem 19 - College Algebra

Question

For the following exercises, find the multiplicative inverse of each matrix, if it exists. $$\left[\begin{array}{rr}0.5 & 1.5 \\ 1 & -0.5\end{array}\right]$$

Step-by-Step Solution

Answer

The inverse of the matrix is \( \begin{bmatrix} \frac{2}{7} & \frac{6}{7} \\ \frac{4}{7} & -\frac{2}{7} \end{bmatrix} \).

Step 1: Determine the Determinant

To find the inverse of a matrix, first calculate its determinant. If the matrix is \( A = \begin{bmatrix} 0.5 & 1.5 \ 1 & -0.5 \end{bmatrix} \), the determinant (det\(A\)) is evaluated as: \( \text{det}(A) = (0.5)(-0.5) - (1.5)(1) \). Calculate this to find \( \text{det}(A) = -0.25 - 1.5 = -1.75 \).

Step 2: Check if Inverse Exists

An inverse exists if and only if the determinant is non-zero. Since \( \text{det}(A) = -1.75 eq 0 \), the matrix does have an inverse.

Step 3: Find the Adjugate Matrix

Compute the adjugate matrix by swapping the elements on the main diagonal and changing the signs of the elements on the other diagonal. So, the adjugate matrix of \( A \), denoted as \( \text{adj}(A) \), is: \[ \text{adj}(A) = \begin{bmatrix} -0.5 & -1.5 \ -1 & 0.5 \end{bmatrix} \].

Step 4: Calculate the Inverse Matrix

The inverse \( A^{-1} \) is given by \( \frac{1}{\text{det}(A)} \times \text{adj}(A) \). Substituting in the determinant and the adjugate matrix, we have: \[ A^{-1} = \frac{1}{-1.75} \times \begin{bmatrix} -0.5 & -1.5 \ -1 & 0.5 \end{bmatrix} = \begin{bmatrix} \frac{2}{7} & \frac{6}{7} \ \frac{4}{7} & -\frac{2}{7} \end{bmatrix} \].

Key Concepts

Determinant of a Matrix

When dealing with matrices, the determinant plays a crucial role in understanding various properties, like invertibility. The determinant for a square matrix provides a single value that helps you understand certain characteristics of the matrix. For a 2x2 matrix, the determinant is computed using a specific formula:

• For a matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated as \( \text{det}(A) = ad - bc \).
• This calculation uses the values from the main diagonal (from top left to bottom right) and subtracts the product of the other diagonal (from top right to bottom left).

As seen in the exercise, finding the determinant is the first step toward establishing if a matrix is invertible. A non-zero determinant indicates that the matrix has an inverse. Meanwhile, a zero determinant means the matrix is singular and not invertible. In our specific scenario, the determinant has been calculated as \(-1.75\), which means an inverse exists.

Adjugate Matrix

The adjugate matrix is an essential component in finding the inverse of a matrix. Once you have the determinant and verify it's non-zero, the next step is constructing the adjugate matrix. This involves some simple transformations:

• First, swap the elements on the main diagonal (diagonal from the top-left corner to the bottom-right corner).
• Next, change the signs of the off-diagonal elements (diagonal from the top-right corner to the bottom-left corner).

Apply these steps to our example matrix \( A = \begin{bmatrix} 0.5 & 1.5 \ 1 & -0.5 \end{bmatrix} \):

• We swap 0.5 and -0.5.
• The signs of 1.5 and 1 change, thus giving us the adjugate matrix \( \text{adj}(A) = \begin{bmatrix} -0.5 & -1.5 \ -1 & 0.5 \end{bmatrix} \).

This adjustment allows the adjugate matrix to effectively "undo" the original matrix when calculating the inverse.

Inverse Matrix Calculation

With the determinant and adjugate matrix in hand, you can calculate the inverse matrix. The inverse of a matrix, if it exists, is crucial because it allows you to solve systems of linear equations among other applications. Here’s how you calculate it:

• The formula for the inverse of a matrix \( A \) is \( A^{-1} = \frac{1}{\text{det}(A)} \times \text{adj}(A) \).
• This means you multiply each element of the adjugate matrix by the reciprocal of the determinant.

For the matrix in the exercise, plugging in our values gives us:

• \( A^{-1} = \frac{1}{-1.75} \times \begin{bmatrix} -0.5 & -1.5 \ -1 & 0.5 \end{bmatrix} \).
• After simplifying, the inverse matrix \( A^{-1} \) becomes \( \begin{bmatrix} \frac{2}{7} & \frac{6}{7} \ \frac{4}{7} & -\frac{2}{7} \end{bmatrix} \).

Calculating the inverse matrix provides a powerful tool for working with matrix operations, granting you the ability to solve problems that involve transformation and linear mappings.