{\left[ {\begin{array}{*{20}{c}}
{\frac{1}{2}}&{\frac{1}{3}}\\\\
{\frac{1}{4}}&{\frac{1}{5}}
\end{array}} \right]^{ - 1}}




To find the inverse of the matrix 

\[
\mathbf{A} = \left[ \begin{array}{cc} \frac{1}{2} & \frac{1}{3} \\ \frac{1}{4} & \frac{1}{5} \end{array} \right],
\]

we can use the formula for the inverse of a 2x2 matrix:

\[
\mathbf{A}^{-1} = \frac{1}{\det(\mathbf{A})} \cdot \left[ \begin{array}{cc} d & -b \\ -c & a \end{array} \right]
\]

where \( \mathbf{A} = \left[ \begin{array}{cc} a & b \\ c & d \end{array} \right] \).

### Step 1: Compute the determinant
For a 2x2 matrix, the determinant is given by:

\[
\det(\mathbf{A}) = ad - bc
\]

Substitute the values:

\[
\det(\mathbf{A}) = \left(\frac{1}{2} \times \frac{1}{5}\right) - \left(\frac{1}{3} \times \frac{1}{4}\right)
\]
\[
= \frac{1}{10} - \frac{1}{12}
\]

To subtract these fractions, find the common denominator:

\[
\det(\mathbf{A}) = \frac{6}{60} - \frac{5}{60} = \frac{1}{60}
\]

### Step 2: Compute the inverse matrix
Now that we have the determinant, which is \( \frac{1}{60} \), the inverse of matrix \( \mathbf{A} \) is:

\[
\mathbf{A}^{-1} = 60 \cdot \left[ \begin{array}{cc} \frac{1}{5} & -\frac{1}{3} \\ -\frac{1}{4} & \frac{1}{2} \end{array} \right]
\]

This simplifies to:

\[
\mathbf{A}^{-1} = \left[ \begin{array}{cc} 12 & -20 \\ -15 & 30 \end{array} \right]
\]

Thus, the inverse of the matrix is:

\[
\boxed{\mathbf{A}^{-1} = \left[ \begin{array}{cc} 12 & -20 \\ -15 & 30 \end{array} \right]}
\]

Would you like further details on any step?

### Related Questions:
1. How do you find the inverse of a 3x3 matrix?
2. What happens if the determinant of a matrix is zero?
3. Can all matrices be inverted?
4. What is the geometric interpretation of a matrix inverse?
5. How does matrix inversion relate to solving linear systems?

### Tip:
When calculating the determinant, be careful with fractions and common denominators to avoid errors.

The query submitted by the user: [ [1/2, 1/3], [1/4, 1/5] ]^(-1)

Learn how to find the inverse of the matrix [1/2, 1/3, 1/4, 1/5] using the formula for the inverse of a 2x2 matrix. This solution walks through computing the determinant and applying it to find the inverse, suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation