Problem 19 - Algebra 2

Question

Find the inverse of each matrix, if it exists. $$ \left[\begin{array}{rr}{4} & {-3} \\ {2} & {7}\end{array}\right] $$

Step-by-Step Solution

Answer

The inverse matrix is \( \begin{bmatrix} \frac{7}{34} & \frac{3}{34} \\ \frac{-1}{17} & \frac{2}{17} \end{bmatrix} \).

Step 1: State the formula for the inverse of a 2x2 matrix

The inverse of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) exists if the determinant is non-zero and is given by the formula \[\begin{bmatrix} a & b \ c & d \end{bmatrix}^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\]

Step 2: Calculate the determinant

First, calculate the determinant of the matrix \( \begin{bmatrix} 4 & -3 \ 2 & 7 \end{bmatrix} \). The determinant \( \Delta \) is calculated as:\[ \Delta = (4)(7) - (-3)(2)\]\[ \Delta = 28 + 6 = 34\]The determinant is 34.

Step 3: Verify if the inverse exists

Since the determinant (\( \Delta = 34 \)) is non-zero, the inverse of the matrix exists.

Step 4: Apply the inverse formula

Using the formula for the inverse of a 2x2 matrix, substitute the values of \( a = 4 \), \( b = -3 \), \( c = 2 \), and \( d = 7 \):\[\begin{bmatrix} 4 & -3 \ 2 & 7 \end{bmatrix}^{-1} = \frac{1}{34} \begin{bmatrix} 7 & 3 \ -2 & 4 \end{bmatrix} = \begin{bmatrix} \frac{7}{34} & \frac{3}{34} \ \frac{-2}{34} & \frac{4}{34} \end{bmatrix}\]Simplify the fractions:\[\begin{bmatrix} \frac{7}{34} & \frac{3}{34} \ \frac{-1}{17} & \frac{2}{17} \end{bmatrix}\]

Step 5: Present the final inverse matrix

The inverse matrix is:\[\begin{bmatrix} \frac{7}{34} & \frac{3}{34} \ \frac{-1}{17} & \frac{2}{17} \end{bmatrix}\]

Key Concepts

2x2 matrix

The 2x2 matrix is one of the simplest types of matrices in mathematics, often used as an introductory example in the study of linear algebra. A matrix is essentially a rectangular array of numbers, and a 2x2 matrix specifically has two rows and two columns.
Its general form is represented as:

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

Here, \( a, b, c, \) and \( d \) are elements of the matrix.
In our given problem, the matrix is \( \begin{bmatrix} 4 & -3 \ 2 & 7 \end{bmatrix} \), where \( a = 4 \), \( b = -3 \), \( c = 2 \), and \( d = 7 \). These numbers can represent various things depending on the problem's context, from coefficients in a system of equations to values in graphics transformations. Understanding a 2x2 matrix is crucial, as it forms the foundation for more complex matrix algebra and operations.

determinant

The determinant is a special number that can be calculated from a square matrix. For a 2x2 matrix, it's a straightforward calculation that provides valuable information about the matrix, such as whether it can be inverted.
The formula for finding the determinant \( \Delta \) of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is:

• \( \Delta = ad - bc \)

In our example, using the matrix \( \begin{bmatrix} 4 & -3 \ 2 & 7 \end{bmatrix} \), the determinant is calculated as follows:

• \( (4)(7) - (-3)(2) = 28 + 6 = 34 \)

A non-zero determinant (as we have here, \( 34 \)) indicates that the matrix has an inverse.
If the determinant were zero, it would mean the matrix is singular, and no inverse exists.

matrix algebra

Matrix algebra involves various operations that can be performed on matrices, such as addition, subtraction, multiplication, and finding the inverse. Finding the inverse of a matrix is particularly useful because it allows us to solve systems of equations and find solutions to linear problems.
To find the inverse of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the formula is:

• \( \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \frac{1}{ad-bc} \)

This formula can only be used when the determinant \( (ad-bc) \) is non-zero.
Using our matrix \( \begin{bmatrix} 4 & -3 \ 2 & 7 \end{bmatrix} \), and given its determinant is 34, the inverse is calculated as:

• \( \frac{1}{34} \begin{bmatrix} 7 & 3 \ -2 & 4 \end{bmatrix} = \begin{bmatrix} \frac{7}{34} & \frac{3}{34} \ \frac{-1}{17} & \frac{2}{17} \end{bmatrix} \)

Matrix algebra is a powerful tool in mathematics, used in a wide range of fields including physics, computer science, and statistics, to solve complex problems and perform data analysis.