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Step 1:I'll solve this step-by-step using proper LaTeX formatting:
Step 2:: Understanding Matrix Inversion for a 2x^2 Matrix
To find the inverse of a 2x2 matrix $$A = \begin{pmatrix} a & b \ c & d \end{pmatrix}$$, we follow a specific process:
Step 3:: Calculate the Determinant
- If $$\det(A) \neq 0$$, the matrix has an inverse
- If \det(A) = 0, the matrix is not invertible
Step 4:: Create the Adjugate Matrix
Create the adjugate matrix by:
Step 5:
Swapping the diagonal elements
Step 6:
\text{adj}(A) = \begin{pmatrix} d & -b \ -c & a \end{pmatrix}
Negating the off-diagonal elements
Step 7:: Calculate the Inverse
A^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix}
The inverse is found by:
Final Answer
For a 2x^2 matrix \begin{pmatrix} a & b \ c & d \end{pmatrix}, its inverse is \frac{1}{ad-bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix} when ad-bc \neq 0.
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