To find a matrix [tex]\(\begin{pmatrix} p & q \\ r & s \end{pmatrix}\)[/tex] that is the inverse of matrix [tex]\(A = \begin{pmatrix} 3 & -1 \\ 5 & -2 \end{pmatrix}\)[/tex], follow these steps:
1. Define Matrix A:
[tex]\[
A = \begin{pmatrix} 3 & -1 \\ 5 & -2 \end{pmatrix}
\][/tex]
2. Recall the Formula for the Inverse of a 2x2 Matrix:
For a general 2x2 matrix [tex]\( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \)[/tex], the inverse is given by:
[tex]\[
A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
\][/tex]
3. Determine the Determinant:
For [tex]\(A = \begin{pmatrix} 3 & -1 \\ 5 & -2 \end{pmatrix}\)[/tex]:
[tex]\[
\text{Det}(A) = (3 \cdot -2) - (-1 \cdot 5) = -6 + 5 = -1
\][/tex]
4. Apply the Inverse Formula:
Using the determinant and swapping the elements in the matrix appropriately:
[tex]\[
A^{-1} = \frac{1}{-1} \begin{pmatrix} -2 & 1 \\ -5 & 3 \end{pmatrix}
\][/tex]
Simplifying, we get:
[tex]\[
A^{-1} = \begin{pmatrix} 2 & -1 \\ 5 & -3 \end{pmatrix}
\][/tex]
5. Check the Result:
Given the true numerical results, the inverse matrix is better seen as:
[tex]\[
A^{-1} = \begin{pmatrix} p & q \\ r & s \end{pmatrix}
\][/tex]
where [tex]\( p \approx 2.000 \)[/tex], [tex]\( q \approx -1.000 \)[/tex], [tex]\( r \approx 5.000 \)[/tex], and [tex]\( s \approx -3.000 \)[/tex].
So, the matrix that is inverse to [tex]\(A\)[/tex] is:
[tex]\[
\begin{pmatrix} \approx 2 & \approx -1 \\ \approx 5 & \approx -3 \end{pmatrix}
\][/tex]
Thus, we can explicitly state [tex]\( p = 2 \)[/tex], [tex]\( q = -1 \)[/tex], [tex]\( r = 5 \)[/tex], and [tex]\( s = -3 \)[/tex] to get:
[tex]\[
\begin{pmatrix} 2 & -1 \\ 5 & -3 \end{pmatrix}
\][/tex]
This is the inverse matrix to [tex]\(A\)[/tex].
