Find λ such that A4 = λA
Given:
\[ A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} \]
Step 1: Find A2
\[ A^2 = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} = \begin{bmatrix} 2 & 2 \\ 2 & 2 \end{bmatrix} = 2A \]
Step 2: Find A3
\[ A^3 = A \cdot A^2 = A \cdot (2A) = 2A^2 = 2(2A) = 4A \]
Step 3: Find A4
\[ A^4 = A \cdot A^3 = A \cdot (4A) = 4A^2 = 4(2A) = 8A \]
Step 4: Compare with λA
\[ A^4 = 8A = \lambda A \Rightarrow \lambda = 8 \]
Final Answer:
\[ \lambda = 8 \]