Question

In Problems 35-44, each matrix is nonsingular. Find the inverse of each matrix. $$ \left[\begin{array}{ll} 2 & 1 \\ 1 & 1 \end{array}\right] $$

Short answer

The inverse matrix is the matrix $$ instead of matrix} = \begin{pmatrix} 1 & -1 \ -1 & 2 \ \text{ }\text{ }\text{ }\text{ }\text{ }\begin{pmatrix}.$$

Step-by-step solution

Identify the matrix

The given matrix is the matrix the matrix $$\begin{pmatrix} 2 & 1 \ 1 & 1 \ \text{ }\text{ }\text{ }\text{ }\text{ }\begin{pmatrix}.

Use the formula for the inverse of a 2x2 matrix

For a 2x2 matrix the matrix the matrix $$\begin{pmatrix} a & b \ c & d \ \text{ }\text{ }\text{ }\text{ }\text{ }\begin{pmatrix}, use \the matrix the matrix $$\text{Inverse} = \frac{1}{\text{ad} - \text{bc}} \begin{pmatrix} d & -b \ -c & a \ \text{ }\text{ }\text{ }\text{ }\text{ }\begin{pmatrix}.$$ Note that the determinant is calculated as $$\text{ad} - \text{bc}$$

Calculate the determinant

Forthe given matrix, the matrix the matrix $$\begin{pmatrix} a = 2, b = 1, c = 1, \text{ }\text{ }\text{ }\text{ }\text{ }\begin{pmatrix}. The determinant can be computed as the matrix $2 the matrix \times 1 - 1 the matrix \times 1\text{ }\text{ }\text{ }\text{ }\text{ } (2 the matrix - 1).

Apply the formula for the inverse

Substitute the values into the inverse formula:the matrix $$\text{Inverse} = \frac{1}{2 - 1} \begin{pmatrix} 1 & -1 \ -1 & 2 \ \text{ }\text{ }\text{ }\text{ }\text{ }\begin{pmatrix}.$$ Simplify the fraction:the matrix $$\text{Inverse} = 1 \begin{pmatrix} 1 & -1 \ -1 & 2 \ \text{ }\text{ }\text{ }\text{ }\text{ }\begin{pmatrix}.$$

Present the final inverse matrix

The inverse of the given matrix is:the matrix $$\begin{pmatrix} 1 & -1 \ -1 & 2 \ \text{ }\text{ }\text{ }\text{ }\text{ }\begin{pmatrix}.$$

Key concepts

2x2 matrix

A 2x2 matrix is a square matrix that has two rows and two columns. In this particular problem, the matrix given is: \( \begin{pmatrix} 2 & 1 \ 1 & 1 \ \text{ }\text{ }\text{ }\text{ }\text{ }\begin{pmatrix} \). This matrix consists of four elements. The elements are arranged as follows: \(\text{}\text{ }\begin{pmatrix} a & b \ c & d \text{ }\text{ }\text{ }\text{ }\begin{pmatrix}\), where \( a = 2, b = 1, c = 1, d = 1 \). Understanding the structure of a 2x2 matrix is essential for finding its inverse.

Determinant

The determinant of a 2x2 matrix is a single value that provides important information about the matrix. It is calculated using the elements of the matrix: \( \text{ad} - \text{bc} \). For the given matrix \( \begin{pmatrix} 2 & 1 \ 1 & 1 \text{ }\text{ }\text{ }\text{ }\begin{pmatrix} \), the determinant is: \( 2 \times 1 - 1 \times 1 = 2 - 1 = 1 \). If the determinant is zero, the matrix does not have an inverse. If the determinant is non-zero, the matrix is called nonsingular and it does have an inverse.

Matrix inverse formula

The formula for finding the inverse of a 2x2 matrix is: \( \text{Inverse} = \frac{1}{\text{ad} - \text{bc}} \begin{pmatrix} d & -b \ -c & a \text{ }\text{ }\text{ }\text{ }\begin{pmatrix} \). This formula uses the determinant \( \text{ad} - \text{bc} \) and rearranges the elements of the original matrix. For our given matrix, substituting the values, we get: \( \text{Inverse} = \frac{1}{1} \begin{pmatrix} 1 & -1 \ -1 & 2 \text{ }\text{ }\text{ }\text{ }\begin{pmatrix} = \begin{pmatrix} 1 & -1 \ -1 & 2 \text{ }\text{ }\text{ }\text{ }\begin{pmatrix} \). This inverse can be used for solving systems of linear equations and other applications.

Nonsingular matrix

A nonsingular matrix is a matrix that has an inverse. For a matrix to be nonsingular, its determinant must be non-zero. In the context of a 2x2 matrix, if \( \text{ad} - \text{bc} eq 0 \), then the matrix is nonsingular. The given matrix \( \begin{pmatrix} 2 & 1 \ 1 & 1 \text{ }\text{ }\text{ }\text{ }\begin{pmatrix} \) has a determinant of 1, which is non-zero. Therefore, it is a nonsingular matrix. The inverse of a nonsingular matrix can provide valuable information and solutions in various mathematical applications, such as solving linear systems and transforming geometric figures.