To find the values of [tex]\( k \)[/tex] and [tex]\( m \)[/tex] for the inverse matrix of [tex]\(\left[\begin{array}{cc}2 & 3 \\ 5 & 9\end{array}\right]\)[/tex] given as [tex]\(\frac{1}{k}\left[\begin{array}{cc}9 & -3 \\ m & 2\end{array}\right]\)[/tex], we proceed with the following steps:
1. Finding the Determinant of the Matrix:
Let [tex]\( A = \left[\begin{array}{cc}2 & 3 \\ 5 & 9\end{array}\right] \)[/tex].
The determinant of matrix [tex]\( A \)[/tex], denoted as [tex]\( \det(A) \)[/tex], is calculated as:
[tex]\[
\det(A) = (2 \cdot 9) - (3 \cdot 5) = 18 - 15 = 3
\][/tex]
Therefore, [tex]\( k = 3 \)[/tex].
2. Finding the Inverse of the Matrix:
The formula for the inverse of a [tex]\( 2 \times 2 \)[/tex] matrix [tex]\( \left[\begin{array}{cc}a & b \\ c & d\end{array}\right] \)[/tex] is:
[tex]\[
A^{-1} = \frac{1}{\det(A)} \left[\begin{array}{cc}d & -b \\ -c & a\end{array}\right]
\][/tex]
Using the elements of our matrix [tex]\( A \)[/tex]:
[tex]\[
A^{-1} = \frac{1}{3} \left[\begin{array}{cc}9 & -3 \\ -5 & 2\end{array}\right]
\][/tex]
3. Comparing the Inverse Matrix:
We are given the form of the inverse matrix as [tex]\(\frac{1}{k}\left[\begin{array}{cc}9 & -3 \\ m & 2\end{array}\right]\)[/tex].
From the calculated inverse, we have:
[tex]\[
A^{-1} = \frac{1}{3} \left[\begin{array}{cc}9 & -3 \\ -5 & 2\end{array}\right] = \left[\begin{array}{cc}\frac{9}{3} & \frac{-3}{3} \\ \frac{-5}{3} & \frac{2}{3}\end{array}\right]
\][/tex]
By comparison, we see:
[tex]\[
k = 3 \quad \text{and} \quad m = -5
\][/tex]
Thus, the values are:
[tex]\[
k = 3 \quad \text{and} \quad m = -5
\][/tex]
