Use technology to find the inverse of the given matrix (when it exists). Round all entries in your answer to two decimal places. (Caution: Because of rounding errors, technology sometimes produces an "inverse" of a singular matrix. These often can be recognized by their huge entries. If the inverse doesn't exist, enter DNE in any cell of the matrix.)

3.57	1.26

−1.03	0

To find the inverse of a 2x2 matrix, we use the following formula:

For a matrix of the form:

\[
A = \begin{pmatrix}
a & b \\
c & d \\
\end{pmatrix}
\]

The inverse \( A^{-1} \), if it exists, is given by:

\[
A^{-1} = \frac{1}{ad - bc} \begin{pmatrix}
d & -b \\
-c & a \\
\end{pmatrix}
\]

where \( ad - bc \) is the determinant of the matrix. If the determinant is 0, the inverse does not exist.

Given the matrix:

\[
A = \begin{pmatrix}
3.57 & 1.26 \\
-1.03 & 0 \\
\end{pmatrix}
\]

First, calculate the determinant:

\[
\text{det}(A) = (3.57 \times 0) - (1.26 \times -1.03) = 0 + 1.2978 = 1.2978
\]

Since the determinant is not zero, the inverse exists.

Now, apply the inverse formula:

\[
A^{-1} = \frac{1}{1.2978} \begin{pmatrix}
0 & -1.26 \\
1.03 & 3.57 \\
\end{pmatrix}
\]

Now, calculate the entries of the inverse matrix:

\[
A^{-1} = \begin{pmatrix}
\frac{0}{1.2978} & \frac{-1.26}{1.2978} \\
\frac{1.03}{1.2978} & \frac{3.57}{1.2978} \\
\end{pmatrix}
= \begin{pmatrix}
0 & -0.97 \\
0.79 & 2.75 \\
\end{pmatrix}
\]

Thus, the inverse matrix (rounded to two decimal places) is:

\[
A^{-1} = \begin{pmatrix}
0 & -0.97 \\
0.79 & 2.75 \\
\end{pmatrix}
\]

Let me know if you need further clarification or details on how the solution was derived.

---

Here are five related questions to explore further:

1. How do you determine whether a matrix is singular or nonsingular?
2. What happens if the determinant of a matrix is zero?
3. Can you apply this method to 3x3 matrices? What changes?
4. What is the significance of the determinant in finding inverses?
5. How are matrix inverses used in solving systems of equations?

**Tip**: Always check the determinant before computing the inverse, as a zero determinant means the matrix is singular, and no inverse exists.

Use technology to find the inverse of the given matrix (when it exists). Round all entries in your answer to two decimal places. (Caution: Because of rounding errors, technology sometimes produces an 'inverse' of a singular matrix. These often can be recognized by their huge entries. If the inverse doesn't exist, enter DNE in any cell of the matrix.)

3.57  1.26
−1.03  0

Learn how to find the inverse of a 2x2 matrix with real-number entries using the determinant. This step-by-step guide covers the inverse matrix formula and determinant calculation for the matrix [[3.57, 1.26], [-1.03, 0]]. The solution shows how to round entries to two decimal places. Suitable for advanced high school and undergraduate students.

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