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.