Step 1: Concept Overview:
This solution utilizes the Cayley-Hamilton theorem, which asserts that a square matrix satisfies its characteristic equation. For a 2x2 matrix, the characteristic equation is given by \(\lambda^2 - \text{tr}(A)\lambda + \det(A) = 0\). Consequently, matrix \(A\) fulfills the equation \(A^2 - \text{tr}(A)A + \det(A)I = 0\).
Step 2: Methodology:
1. Determine the trace (\(\text{tr}(A) = a+d\)) and the determinant (\(\det(A) = ad-bc\)) from the provided data.
2. Formulate the Cayley-Hamilton equation for matrix \(A\).
3. Algebraically manipulate the equation to derive an expression for \(A^3\).
Step 3: Step-by-Step Solution:
We are given the 2x2 matrix \(A\) with the following properties:
\(\text{tr}(A) = a+d = 1\)
\(\det(A) = ad-bc = 1\)
According to the Cayley-Hamilton theorem, for a 2x2 matrix:
\[ A^2 - (\text{tr}(A))A + (\det(A))I = 0 \]
Substituting the provided values:
\[ A^2 - (1)A + (1)I = 0 \]
\[ A^2 - A + I = 0 \]
Therefore, \(A^2\) can be expressed as:
\[ A^2 = A - I \]
To determine \(A^3\), multiply the equation by \(A\):
\[ A(A^2) = A(A - I) \]
\[ A^3 = A^2 - AI \]
\[ A^3 = A^2 - A \]
Substituting the expression for \(A^2\) into the equation:
\[ A^3 = (A - I) - A \]
\[ A^3 = A - I - A \]
\[ A^3 = -I \]
Step 4: Conclusion:
The resulting matrix \(A^3\) is \(-I\).