Problem 32 - A Graphical Approach to College Algebra

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

Matrix \( A \) is invertible, and the inverse is given by the calculated expression.

Step 1: Determine Matrix Size and Type

Matrix \( A \) is a \( 2 \times 2 \) matrix given by: \[ A=\begin{bmatrix} \frac{2}{3} & 0.7 \ 22 & \sqrt{3} \end{bmatrix} \] Since it is a square matrix, it is potentially invertible if its determinant is non-zero.

Step 2: Calculate the Determinant

The determinant of a \( 2 \times 2 \) matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is calculated as \( ad - bc \).For matrix \( A \), substitute the values:\[ \text{det}(A) = \left(\frac{2}{3}\right)(\sqrt{3}) - (0.7)(22) \] Calculate this to find the determinant.

Step 3: Simplify Determinant Calculation

First, compute the terms separately:\( \frac{2}{3} \times \sqrt{3} = \frac{2\sqrt{3}}{3} \)\( 0.7 \times 22 = 15.4 \)Thus, the determinant is:\[ \text{det}(A) = \frac{2\sqrt{3}}{3} - 15.4 \] Since \( \text{det}(A) = \frac{2\sqrt{3}}{3} - 15.4 eq 0 \), the matrix is invertible.

Step 4: Apply Formula for Inverse of 2x2 Matrix

The inverse of a \( 2 \times 2 \) matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is given by:\[ A^{-1} = \frac{1}{ad - bc}\begin{bmatrix} d & -b \ -c & a \end{bmatrix} \] Substitute the values from matrix \( A \) and its determinant:\[ A^{-1} = \frac{1}{\frac{2\sqrt{3}}{3} - 15.4}\begin{bmatrix} \sqrt{3} & -0.7 \ -22 & \frac{2}{3} \end{bmatrix} \]

Step 5: Calculate Inverse Elements

Compute each element for the inverse matrix:First, compute the scalar \( \frac{1}{\frac{2\sqrt{3}}{3} - 15.4} \).Then, multiply each element inside the matrix by this scalar:\[ A^{-1} = \left( \frac{1}{\frac{2\sqrt{3}}{3} - 15.4} \right) \begin{bmatrix} \sqrt{3} & -0.7 \ -22 & \frac{2}{3} \end{bmatrix} \]

Key Concepts

Determinant Calculation

To find out if a square matrix is invertible, we need to calculate its determinant. This is particularly straightforward for a 2x2 matrix, a commonly encountered type in many math exercises. For a 2x2 matrix represented as \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant \( \text{det}(A) \) is calculated using the formula: \( ad - bc \).

For matrix \( A \) in the exercise, follow these simple steps:

• Multiply the top left element by the bottom right element and the top right by the bottom left, i.e., \( (\frac{2}{3})(\sqrt{3}) - (0.7)(22) \).
• Simplify this expression to get the determinant.
• If the calculated determinant is non-zero, the matrix is invertible.

This makes the determinant calculation an important first step towards finding a matrix inverse.

2x2 Matrix

A 2x2 matrix is a simple form of a matrix containing two rows and two columns. Expressed in this form \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), it's one of the most fundamental types of matrices you'll encounter.

The key properties of 2x2 matrices make them very manageable for computing inverses and determinants, which is why they serve as a great starting point for learning linear algebra.

• Each element in the 2x2 matrix contributes to the computation of the determinant and the inverse.
• Their size makes manual calculations feasible even without advanced software or a calculator.

The simplicity of 2x2 matrices helps students grasp bigger concepts in the world of matrices and linear algebra.

Invertible Matrix

An invertible matrix, sometimes called a non-singular matrix, is one that possesses an inverse.

For matrices to have inverses, several conditions must be met, especially the determinant must not be zero. In this exercise, the matrix \( A \) was found to have a non-zero determinant, confirming it is indeed invertible.

• The inverse of a 2x2 matrix can be found using the formula: \( A^{-1} = \frac{1}{ad-bc}\begin{bmatrix} d & -b \ -c & a \end{bmatrix} \), where \( ad-bc \) is the determinant.
• The concept of an inverse is pivotal in solving systems of linear equations and performing matrix algebra.

Once confirmed invertible, the calculation follows the straightforward pattern dictated by its determinant. Understanding this process makes broader matrix operations much simpler to tackle.