https://file.aiquickdraw.com/mathbot/file/1729448408372102uximg.jpg

The image presents the problem of finding the inverse of the matrix:

\[
\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}^{-1}
\]

We need to calculate the inverse of this matrix.

To find the inverse of a 2x2 matrix:

\[
\text{For a matrix } A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \text{ the inverse is given by } A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix},
\]

provided that \( ad - bc \neq 0 \), which is the determinant.

For the matrix:

\[
A = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}
\]

- \( a = 1 \), \( b = 2 \), \( c = 0 \), and \( d = 1 \).
- The determinant \( ad - bc = 1 \times 1 - 0 \times 2 = 1 \).

Thus, the inverse is:

\[
A^{-1} = \frac{1}{1} \begin{pmatrix} 1 & -2 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & -2 \\ 0 & 1 \end{pmatrix}.
\]

Now, checking the options, the correct answer is:

\[
\boxed{a. \begin{pmatrix} 1 & -2 \\ 0 & 1 \end{pmatrix}}.
\]

Would you like further clarification on this process or any additional details? 

Here are 5 related questions for further practice:
1. What is the determinant of a 2x2 matrix?
2. How do you calculate the inverse of a 3x3 matrix?
3. If the determinant of a matrix is 0, what does that imply about its inverse?
4. Can every square matrix be inverted? Why or why not?
5. What is the relationship between a matrix and its inverse?

**Tip:** To verify if a matrix is its inverse, multiply the matrix by its inverse. The result should be the identity matrix.

Find the inverse of the matrix A = [[1, 2], [0, 1]]. Choose the correct option from the following answers.

This problem involves calculating the inverse of a 2x2 matrix A = [[1, 2], [0, 1]]. Learn how to use the inverse matrix formula and determinant calculations to solve similar problems. Suitable for students in Grades 10-12.

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