https://file.aiquickdraw.com/mathbot/file/1727097859479n684oq4j.jpg

The question in the image asks to find the inverse of matrix \( F \), where:

\[
F = \begin{bmatrix} 2 & -4 \\ -1 & 1 \end{bmatrix}
\]

To find the inverse of a 2x2 matrix \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), we use the formula:

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

where \( \text{det}(F) \) is the determinant of matrix \( F \), calculated as:

\[
\text{det}(F) = ad - bc
\]

### Step 1: Calculate the determinant

For the matrix \( F = \begin{bmatrix} 2 & -4 \\ -1 & 1 \end{bmatrix} \),

\[
\text{det}(F) = (2)(1) - (-4)(-1) = 2 - 4 = -2
\]

### Step 2: Apply the formula for the inverse

Now, the inverse of \( F \) is:

\[
F^{-1} = \frac{1}{-2} \begin{bmatrix} 1 & 4 \\ 1 & 2 \end{bmatrix} = \begin{bmatrix} -\frac{1}{2} & -2 \\ -\frac{1}{2} & -1 \end{bmatrix}
\]

Thus, the inverse of matrix \( F \) is:

\[
F^{-1} = \begin{bmatrix} -\frac{1}{2} & -2 \\ -\frac{1}{2} & -1 \end{bmatrix}
\]

Would you like more details on the process or have any questions?

Here are 5 related questions to explore:
1. How do you calculate the inverse of a 3x3 matrix?
2. What happens if the determinant of a matrix is zero?
3. How can you verify that the inverse of a matrix is correct?
4. What are the properties of invertible matrices?
5. Can non-square matrices have inverses?

**Tip:** Always check the determinant before calculating the inverse—if it's zero, the matrix does not have an inverse!

Tentukan invers dari matriks F = [2 -4; -1 1] !

Learn how to find the inverse of the 2x2 matrix F = [2 -4; -1 1] using the matrix inverse formula. This problem involves calculating the determinant and applying matrix inversion, suitable for students in Grades 10-12.

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