Problem 31 - A Graphical Approach to Precalculus with Limits

Question

For each matrix, find \(A^{-1}\) if it exists. $$A=\left[\begin{array}{cc} \sqrt{2} & 0.5 \\ -17 & \frac{1}{2} \end{array}\right]$$

Step-by-Step Solution

Answer

The inverse of matrix \( A \) is calculated using the inverse formula, resulting in a matrix with terms divided by the non-zero determinant.

Step 1: Identify the matrix

We have the matrix \( A \) given by: \[ A = \begin{bmatrix} \sqrt{2} & 0.5 \ -17 & \frac{1}{2} \end{bmatrix} \]

Step 2: Calculate the Determinant of A

The determinant of a \( 2 \times 2 \) matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is calculated using the formula:\( \text{det}(A) = ad - bc \).For our matrix:- \( a = \sqrt{2} \)- \( b = 0.5 \)- \( c = -17 \)- \( d = \frac{1}{2} \)Calculate:\( \text{det}(A) = (\sqrt{2})(0.5) - (0.5)(-17) = 0.5\sqrt{2} + 8.5 \)

Step 3: Check if Inverse Exists

An inverse exists if the determinant is non-zero. From step 2, we have:\( \text{det}(A) = 0.5\sqrt{2} + 8.5 \).This expression is non-zero, so the inverse exists.

Step 4: Use the Formula for Inverse of a 2x2 Matrix

The inverse of a \( 2 \times 2 \) matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is given by:\[ A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \]Substitute \( a = \sqrt{2} \), \( b = 0.5 \), \( c = -17 \), \( d = \frac{1}{2} \):\[ A^{-1} = \frac{1}{0.5\sqrt{2} + 8.5} \begin{bmatrix} 0.5 & -0.5 \ 17 & \sqrt{2} \end{bmatrix} \]

Step 5: Calculate the Inverse Matrix

First calculate the reciprocal of the determinant:\[ \frac{1}{0.5\sqrt{2} + 8.5} \]Now apply it to the matrix:\[ A^{-1} = \begin{bmatrix} \frac{0.5}{0.5\sqrt{2} + 8.5} & \frac{-0.5}{0.5\sqrt{2} + 8.5} \ \frac{17}{0.5\sqrt{2} + 8.5} & \frac{\sqrt{2}}{0.5\sqrt{2} + 8.5} \end{bmatrix} \]

Key Concepts

Determinant Calculation

To determine if a matrix has an inverse, we first need to calculate its determinant. The determinant of a 2x2 matrix is a crucial value that tells us about the matrix's invertibility. For a general matrix \[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \]the determinant is calculated with the formula:\[ \text{det}(A) = ad - bc \] This formula involves multiplying the top left entry by the bottom right (\( ad \)) and subtracting the product of the top right entry by the bottom left (\( bc \)).

In our example, the matrix \[ A = \begin{bmatrix} \sqrt{2} & 0.5 \ -17 & \frac{1}{2} \end{bmatrix} \]uses the entries \( a = \sqrt{2}, b = 0.5, c = -17, \) and \( d = \frac{1}{2} \). Plugging these values into the determinant formula, we get:\[ \text{det}(A) = (\sqrt{2})(0.5) - (0.5)(-17) = 0.5\sqrt{2} + 8.5 \] This result is crucial because it informs us of the matrix's properties, like its invertibility.

Inverse of a 2x2 Matrix

Once we have the determinant, we can move on to finding the inverse of the 2x2 matrix. The inverse only exists if the determinant is non-zero. The formula for the inverse of a matrix \[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \] is given by:\[ A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \] Here’s the step-by-step process:

• Calculate the determinant \( ad - bc \).
• If the determinant is non-zero, compute \( \frac{1}{ad - bc} \).
• Plug these values into the inverse formula matrix positions: swap \( a \) and \( d \), change signs of \( b \) and \( c \).

For our specific matrix, the entries are \( a = \sqrt{2} \), \( b = 0.5 \), \( c = -17 \), and \( d = \frac{1}{2} \). Their inverse positions and signs are then used in:\[ A^{-1} = \frac{1}{0.5\sqrt{2} + 8.5} \begin{bmatrix} 0.5 & -0.5 \ 17 & \sqrt{2} \end{bmatrix} \] It’s essential to align these new values correctly to make sure results are accurate.

Non-zero Determinant Condition

A determinant can tell us whether a matrix is invertible or not. Specifically, a matrix is only invertible if its determinant is not zero. This is known as the non-zero determinant condition. Such a condition ensures that the reciprocal of the determinant \( \frac{1}{ad - bc} \) is defined, allowing the inverse calculation to proceed.

If the determinant is zero, \( \frac{1}{0} \) is undefined, and thus, the matrix has no inverse. In practical terms, this means:

• Always confirm the determinant isn't zero before calculating an inverse.
• Understand that if the determinant is zero, the matrix represents a transformation that squashes dimensions, losing uniqueness.

In the provided exercise example, the determinant was calculated as \( 0.5\sqrt{2} + 8.5 \), which is clearly non-zero. Consequently, this non-zero condition assured us of the possibility to continue with finding the inverse matrix. It's a pivotal step confirming that the matrix transformation maintains its distinctiveness and invertibility.