Question:
Find the inverse of the matrix:
\[ \begin{bmatrix} 2 & -3 \\ 3 & 2 \end{bmatrix} \]
Concept:
For a matrix of the form:
\[ \begin{bmatrix} a & -b \\ b & a \end{bmatrix} \]
Its inverse is:
\[ \frac{1}{a^2 + b^2} \begin{bmatrix} a & b \\ – b & a \end{bmatrix} \]
Solution:
Here, \( a = 2 \), \( b = 3 \)
\[ a^2 + b^2 = 4 + 9 = 13 \]
So inverse is:
\[ \frac{1}{13} \begin{bmatrix} 2 & 3 \\ -3 & 2 \end{bmatrix} \]
—Final Answer:
\[ \boxed{ \frac{1}{13} \begin{bmatrix} 2 & 3 \\ -3 & 2 \end{bmatrix} } \]