Problem 15 - A Graphical Approach to Precalculus with Limits

Question

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{ll} 0.6 & 0.2 \\ 0.5 & 0.1 \end{array}\right]$$

Step-by-Step Solution

Answer

The inverse of matrix \(A\) is \(\begin{bmatrix} -2.5 & 5 \\ 12.5 & -15 \end{bmatrix}\).

Step 1: Find the Determinant

The first step to finding the inverse of a 2x2 matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\) is to compute its determinant \(\det(A)\). For matrix \(A = \begin{bmatrix} 0.6 & 0.2 \ 0.5 & 0.1 \end{bmatrix}\), the determinant is calculated as follows:\[\text{det}(A) = ad - bc = (0.6)(0.1) - (0.2)(0.5) = 0.06 - 0.1 = -0.04\]

Step 2: Check if the Determinant is Zero

The inverse of a matrix exists if and only if its determinant is not zero. In this case, \(\det(A) = -0.04\), which is not zero. Therefore, the inverse of matrix \(A\) exists.

Step 3: Compute the Inverse Matrix

The inverse of a 2x2 matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\) is given by:\[A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\]Substitute the given values into the formula:\[A^{-1} = \frac{1}{-0.04} \begin{bmatrix} 0.1 & -0.2 \ -0.5 & 0.6 \end{bmatrix} = \begin{bmatrix} -2.5 & 5 \ 12.5 & -15 \end{bmatrix}\]So, the inverse of matrix \(A\) is \(\begin{bmatrix} -2.5 & 5 \ 12.5 & -15 \end{bmatrix}\).

Key Concepts

Determinant

The determinant is a special number that you can calculate from a square matrix. It plays a crucial role in matrix operations, especially when you need to find the inverse of a matrix.
For a 2x2 matrix \[A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\]its determinant is computed by the formula:\[\text{det}(A) = ad - bc\]This determinant helps determine if a matrix is invertible.
If the determinant is zero, the matrix does not have an inverse. In the given exercise, the determinant is -0.04, which is non-zero. Therefore, an inverse exists for the matrix.

2x2 Matrix

A 2x2 matrix is one of the simplest forms of matrices in linear algebra. It is composed of two rows and two columns.
Here’s an example format of a 2x2 matrix:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]Each element, like \(a\), \(b\), \(c\), and \(d\), represents a particular entry of the matrix.
Despite its simplicity, this type of matrix is fundamental in mathematics and engineering due to its straightforward operations and foundational learning potential.
Understanding the basic operations on a 2x2 matrix such as finding the determinant and inverses is essential for advancing to more complex topics in matrix algebra.

Inverse Calculation

Finding the inverse of a matrix is like finding a reciprocal in numbers.
The inverse operation is important in solving matrix equations and finding linear transformations. For a 2x2 matrix:\[A = \begin{bmatrix} a & b \ c & d \end{bmatrix},\]the formula for its inverse, denoted as \(A^{-1}\), is:\[A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}.\]This formula involves swapping the positions of \(a\) and \(d\), changing the signs of \(b\) and \(c\), and dividing by the determinant.
In our exercise, this provides us an inverse of:\[\begin{bmatrix} -2.5 & 5 \ 12.5 & -15 \end{bmatrix}.\]This process is straightforward yet powerful in solving systems of linear equations.

Matrix Algebra

Matrix algebra is a branch of mathematics that deals with matrix operations.
It encompasses various rules and operations applicable to matrices, such as addition, subtraction, multiplication, and inversion.
Understanding matrix algebra is crucial, as it forms the basis for more complicated calculations in linear transformations and systems of equations.

• Addition and Subtraction: Matrices can be added or subtracted if they have the same dimensions.
• Multiplication: The multiplication of matrices involves rows and columns, requiring careful alignment.
• Inversion: Finding the inverse is possible only for square matrices (e.g., 2x2).

These operations make matrix algebra a powerful tool in various applications, including computer graphics, physics, and economics, where systems of linear equations are prevalent.