Question

Given a square matrix \(A\) where \(A^2 + I = 2A\), find \(A^9\).

• A. \(8A^2 - 7I\)
• B. \(9A + 8I\)
• C. \(9A - 8I\)
• D. \(8A^2 + 7I\)

Hint:

Leverage initial equations to simplify matrix power calculations, avoiding extensive computational errors.

Answer 1

The Correct Option is C

Step 1: Utilizing the matrix equation.
From \(A^2 + I = 2A\), we can express \(A^2 = 2A - I\). 

Step 2: Calculating higher powers of \(A\).
\[ A^3 = A \cdot A^2 = A(2A - I) = 2A^2 - A = 2(2A - I) - A = 4A - 2I - A = 3A - 2I \] \[ A^4 = A \cdot A^3 = A(3A - 2I) = 3A^2 - 2A = 3(2A - I) - 2A = 6A - 3I - 2A = 4A - 3I \] Continuing this way, calculate up to \(A^9\). 

Step 3: Expressing \(A^9\) in terms of \(A\) and \(I\).
Through repeated squaring and multiplication, we find: \[ A^9 = 9A - 8I \]