To determine the correct inverse matrix expression for a $2 \times 2$ matrix, we need to recall the formula for the inverse of a $2 \times 2$ matrix. For a matrix:

$\begin{bmatrix} a & b \\ c & d \end{bmatrix}$

The inverse exists if the determinant $\text{det} = ad - bc$ is non-zero. The inverse is given by:

$\frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$

Now, let's compare this formula with the given options in the image:

• Option (A): $\frac{1}{ad - bc} \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ — Incorrect, as it simply repeats the original matrix.
• Option (B): $\frac{1}{ad - bc} \begin{bmatrix} -a & -b \\ -c & -d \end{bmatrix}$ — Incorrect, as this matrix just negates the original elements.
• Option (C): $\frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$ — Correct, as it matches the formula for the inverse.
• Option (D): $\frac{1}{ad - bc} \begin{bmatrix} d & c \\ b & a \end{bmatrix}$ — Incorrect, as it doesn’t match the required structure for the inverse.

Therefore, the correct answer for the first question is Option C.

For the second question, which asks about the determinant of a matrix:

• If the determinant is less than zero, that means it's not zero, and thus the inverse does exist. This eliminates the possibility that "the inverse matrix doesn't exist."

Thus, the correct answer for the second question is Option A: "The inverse matrix exists."

Would you like more details on these solutions, or do you have any other questions?

Here are five related questions for further exploration:

1. How do we calculate the determinant for higher-order matrices, like $3 \times 3$?
2. What are some applications of inverse matrices in real-world problems?
3. How does the inverse of a matrix change if we modify one of its elements?
4. What happens if a matrix has a determinant of zero?
5. How are inverse matrices used in solving systems of linear equations?

Tip: Always check the determinant of a matrix before attempting to find its inverse. If the determinant is zero, the inverse does not exist.