https://file.aiquickdraw.com/mathbot/file/17285095365778yjjvga7.jpg

To find the inverse of a 2x2 matrix \( C \), we can use the following formula for the inverse of a matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \):

\[
A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
\]

In this case, the matrix \( C \) is given as:

\[
C = \begin{pmatrix} -2 & 3 \\ -2 & 8 \end{pmatrix}
\]

Let's apply the formula step by step:

1. **Compute the determinant**:  
   The determinant of a 2x2 matrix is calculated as:

   \[
   \text{det}(C) = (-2)(8) - (3)(-2) = -16 + 6 = -10
   \]

2. **Use the inverse formula**:  
   Now, use the formula for the inverse:

   \[
   C^{-1} = \frac{1}{-10} \begin{pmatrix} 8 & -3 \\ 2 & -2 \end{pmatrix}
   \]

   Multiplying by \( \frac{1}{-10} \):

   \[
   C^{-1} = \begin{pmatrix} \frac{8}{-10} & \frac{-3}{-10} \\ \frac{2}{-10} & \frac{-2}{-10} \end{pmatrix}
   \]

3. **Simplify the fractions**:  
   Simplifying the fractions:

   \[
   C^{-1} = \begin{pmatrix} \frac{-4}{5} & \frac{3}{10} \\ \frac{1}{5} & \frac{-1}{5} \end{pmatrix}
   \]

So, the inverse matrix is:

\[
C^{-1} = \begin{pmatrix} \frac{-4}{5} & \frac{3}{10} \\ \frac{1}{5} & \frac{-1}{5} \end{pmatrix}
\]

Would you like more details or have any questions about the solution?

Here are some related questions for practice:
1. How do you compute the inverse of a 3x3 matrix?
2. What happens if the determinant of a matrix is zero?
3. How can matrix inverses be used to solve systems of equations?
4. What is the relationship between the determinant and the invertibility of a matrix?
5. How do you interpret the inverse of a transformation matrix in geometry?

**Tip:** The inverse of a matrix exists only if the determinant is non-zero!

Determine the inverse of the matrix C = [ [-2, 3], [-2, 8] ]

Learn how to find the inverse of a 2x2 matrix C = [ [-2, 3], [-2, 8] ] with a step-by-step solution. The process involves calculating the determinant and applying the matrix inversion formula. This problem covers key linear algebra concepts and is suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation