To find the inverse of the given [tex]\(2 \times 2\)[/tex] matrix [tex]\(A\)[/tex], it's essential to take note of the steps involved in calculating it. The given matrix [tex]\(A\)[/tex] is:
[tex]\[
A = \begin{pmatrix}
-9 & 34 \\
5 & -19
\end{pmatrix}
\][/tex]
The first step is to find the determinant of the matrix [tex]\(A\)[/tex]. For a [tex]\(2 \times 2\)[/tex] matrix:
[tex]\[
A = \begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\][/tex]
the determinant is calculated as:
[tex]\[
\text{det}(A) = ad - bc
\][/tex]
Substitute the values from matrix [tex]\(A\)[/tex]:
[tex]\[
\text{det}(A) = (-9)(-19) - (34)(5) = 171 - 170 = 1
\][/tex]
The determinant of the matrix is [tex]\(1\)[/tex].
Since the determinant is not zero, the matrix [tex]\(A\)[/tex] is invertible. Next, we need to find the inverse of the matrix. The formula to find the inverse of a [tex]\(2 \times 2\)[/tex] matrix [tex]\(A\)[/tex] is given by:
[tex]\[
A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix}
d & -b \\
-c & a
\end{pmatrix}
\][/tex]
Given [tex]\(\text{det}(A) = 1\)[/tex], substitute the values into the formula:
[tex]\[
A^{-1} = \begin{pmatrix}
-19 & -34 \\
-5 & -9
\end{pmatrix}
\][/tex]
Thus, the inverse of the given matrix is:
[tex]\[
\begin{pmatrix}
-19 & -34 \\
-5 & -9
\end{pmatrix}
\][/tex]
