Let A be a 2 × 2 matrix such that trace(A) = 5 and A² - 5A + 6I = O. What are the possible values for the elements on the principal diagonal of an equivalent diagonal matrix?

Let A be a 2 × 2 matrix such that trace(A) = 5 and A² - 5A + 6I = O. What are the possible values for the elements on the principal diagonal of an equivalent diagonal matrix?

• A. 2 and 3
• B. 1 and 4
• C. 0 and 5
• D. -2 and -3

Answer: A) 2 and 3

Explanation: The characteristic equation of the matrix is given by x² - 5x + 6 = 0, whose roots (eigenvalues) are 2 and 3. The sum of the roots is 5, which matches the trace. The diagonal entries of the equivalent diagonal matrix are its eigenvalues, 2 and 3.