Problem 32 - A Graphical Approach to Precalculus with Limits

Question

For each matrix, find \(A^{-1}\) if it exists. $$A=\left[\begin{array}{ll} \frac{2}{3} & 0.7 \\ 22 & \sqrt{3} \end{array}\right]$$

Step-by-Step Solution

Answer

The inverse of matrix \( A \) is \( \left[\begin{array}{cc} -0.122 & 0.049 \\ 1.545 & -0.047 \end{array}\right] \).

Step 1: Calculate the Determinant

First, calculate the determinant of matrix \( A \). For a 2x2 matrix given by \( A = \left[\begin{array}{cc} a & b \ c & d \end{array}\right] \), the determinant \( \det(A) \) is calculated as \( ad - bc \). For matrix \( A = \left[\begin{array}{cc} \frac{2}{3} & 0.7 \ 22 & \sqrt{3} \end{array}\right] \), we find:\[det(A) = \left( \frac{2}{3} \right)(\sqrt{3}) - (0.7)(22) = \frac{2\sqrt{3}}{3} - 15.4 \]

Step 2: Evaluate Determinant

Now, compute \( \frac{2\sqrt{3}}{3} \) and subtract 15.4. If the determinant is not zero, the inverse exists.Approximating \( \sqrt{3} \approx 1.732 \), we calculate:\[\frac{2\cdot1.732}{3} \approx \frac{3.464}{3} \approx 1.155\]So, the determinant becomes:\[1.155 - 15.4 \approx -14.245\]Since \( -14.245 eq 0 \), the inverse exists.

Step 3: Setup the Inverse Matrix Formula

Recall that the inverse of a 2x2 matrix \( A \) is given by:\[A^{-1} = \frac{1}{\det(A)} \left[\begin{array}{cc} d & -b \ -c & a \end{array}\right]\]Substitute the values from matrix \( A \) to get:\[A^{-1} = \frac{1}{-14.245} \left[\begin{array}{cc} \sqrt{3} & -0.7 \ -22 & \frac{2}{3} \end{array}\right]\]

Step 4: Calculate the Inverse Matrix

Multiply each element of the matrix by \( \frac{1}{-14.245} \) to find the inverse:\[A^{-1} = \left[\begin{array}{cc} \frac{-\sqrt{3}}{14.245} & \frac{0.7}{14.245} \ \frac{22}{14.245} & \frac{-2}{3 \times 14.245} \end{array}\right]\]Simplifying, using the approximation \( \sqrt{3} \approx 1.732 \):\[A^{-1} \approx \left[\begin{array}{cc} -0.122 & 0.049 \ 1.545 & -0.047 \end{array}\right]\]

Step 5: Verify the Result

Confirm by multiplying the original matrix \( A \) by the computed inverse \( A^{-1} \). Ensure the result is approximately the identity matrix: \( A \cdot A^{-1} \approx I \). The calculations should yield the identity matrix, confirming the correctness of \( A^{-1} \).

Key Concepts

Determinant Calculation

Calculating the determinant of a matrix is a crucial step in finding the inverse of a matrix. The determinant can be thought of as a special number that helps us understand certain properties of a matrix, such as whether the matrix is invertible or not.
For a 2x2 matrix, calculating the determinant is quite straightforward. If your matrix is given as:

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

The determinant \( \det(A) \) is calculated using the formula: \( ad - bc \).
In our given problem:

• \( a = \frac{2}{3} \)
• \( b = 0.7 \)
• \( c = 22 \)
• \( d = \sqrt{3} \)

The formula becomes: \( \det(A) = \left( \frac{2}{3} \right)\sqrt{3} - (0.7)(22) \).
This calculation helps to determine if the matrix can have an inverse. If the determinant is zero, the matrix does not have an inverse.

Inverse Matrix Formula

Once you've calculated the determinant and confirmed it's not zero, the next step is to find the inverse of the matrix. There is a specific formula for the inverse of a 2x2 matrix. If you have matrix \( A \):

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

The inverse \( A^{-1} \) is given by the formula:

• \( A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \)

In this formula:

• \( d \) and \( a \) are swapped
• \( b \) and \( c \) change their signs

The structure of the formula helps in reversing the effect of applying matrix \( A \), essentially "undoing" its operation.
In our exercise, you substitute the elements of the original matrix and divide by the determinant to form the inverse matrix.

Identity Matrix Verification

Verification of the inverse matrix is an essential part of the solving process. You must multiply the original matrix by its calculated inverse. The goal is to confirm that the product is the identity matrix. For a 2x2 matrix, the identity matrix looks like this:

• \( I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \)

When you multiply any matrix by its inverse, the result should approximate this identity matrix.
In the case of our problem, we take:

• Original matrix \( A \)
• Inverse matrix \( A^{-1} \) that we calculated

Perform the multiplication \( A \cdot A^{-1} \).
If this product closely matches the identity matrix, it confirms the calculation of the inverse was correct. This check is crucial to ensure that every step was done properly and the inverse matrix will work right. This is especially useful in applications like solving systems of equations.