Problem 11 - A Graphical Approach to College Algebra

Question

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{rr} -1 & -2 \\ 3 & 4 \end{array}\right]$$

Step-by-Step Solution

Answer

The inverse of matrix A is \( A^{-1} = \begin{bmatrix} 2 & 1 \\ -\frac{3}{2} & -\frac{1}{2} \end{bmatrix} \).

Step 1: Determine the Determinant

To find the inverse of matrix \( A \), we first need to calculate its determinant, denoted as \( \text{det}(A) \). For a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is computed as \( \text{det}(A) = ad - bc \). For matrix \( A = \begin{bmatrix} -1 & -2 \ 3 & 4 \end{bmatrix} \), we use \( a = -1 \), \( b = -2 \), \( c = 3 \), and \( d = 4 \). The determinant is thus \( (-1)(4) - (-2)(3) = -4 + 6 = 2 \). Therefore, \( \text{det}(A) = 2 \).

Step 2: Check for Invertibility

A matrix is invertible if and only if its determinant is non-zero. In our case, \( \text{det}(A) = 2 \), which is non-zero. Thus, matrix \( A \) is invertible.

Step 3: Compute the Inverse Matrix

For a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the inverse \( A^{-1} \) is given by:\[ A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \]Substitute the values from matrix \( A = \begin{bmatrix} -1 & -2 \ 3 & 4 \end{bmatrix} \):\( a = -1 \), \( b = -2 \), \( c = 3 \), and \( d = 4 \). Thus, the inverse matrix is:\[ A^{-1} = \frac{1}{2} \begin{bmatrix} 4 & 2 \ -3 & -1 \end{bmatrix} = \begin{bmatrix} 2 & 1 \ -\frac{3}{2} & -\frac{1}{2} \end{bmatrix} \].

Key Concepts

2x2 Matrices

2x2 matrices are fundamental elements in linear algebra that consist of four elements arranged in two rows and two columns. These matrices are quite common in various mathematical computations and are simple enough to handle manually, which is beneficial for understanding basic concepts like matrix inversion and determinant calculation. In a general form, a 2x2 matrix is represented as:

• \[A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\]

Each element within the matrix can be a number, and the matrix can represent a variety of linear transformations. Understanding how to use these matrices in calculations is crucial for more advanced topics in mathematics, like solving systems of equations. The operations performed on 2x2 matrices, especially matrix inversion, require a basic understanding of their structure and composition.

Determinant Calculation

The determinant of a 2x2 matrix plays a crucial role in determining whether a matrix has an inverse. It is a single value that provides valuable properties of the matrix, such as its invertibility. For a matrix:

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

The formula to calculate the determinant, denoted as \(\text{det}(A)\), is:

• \(\text{det}(A) = ad - bc\)

To determine if a 2x2 matrix is invertible, check if the determinant is not zero. In our example, the determinant is calculated as:- \((-1) \times 4 - (-2) \times 3 = -4 + 6 = 2\)This positive determinant confirms that the given matrix is invertible, allowing us to proceed with finding its inverse.

Matrix Algebra

Matrix algebra involves various operations that can be performed on matrices, such as addition, subtraction, and multiplication. One key operation is calculating an inverse, especially for solving systems of equations or transforming coordinate systems. To find the inverse of a matrix, especially a 2x2 matrix, use the formula:

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

The inverse exists only if the determinant is non-zero, ensuring that the matrix has full rank. In our case, with matrix \(A = \begin{bmatrix} -1 & -2 \ 3 & 4 \end{bmatrix}\), its inverse is calculated by plugging the values into the inverse formula, resulting in:- \(A^{-1} = \begin{bmatrix} 2 & 1 \ -\frac{3}{2} & -\frac{1}{2} \end{bmatrix}\)This operation is crucial in various applications, including computer graphics, system modeling, and more, showcasing the power of matrix algebra in handling different problems.