Question

Let $A$ be an $n \times n$ matrix such that $A^3 = A$. Then which one of the following is true?

• A. $A$ must be an identity matrix.
• B. $A^2$ must be an identity matrix.
• C. $A$ is invertible.
• D. $A$ is diagonalisable.

Hint:

Any matrix satisfying an equation of the form $A^k = A$ (for $k > 1$) is diagonalizable because the polynomial $x^k - x = 0$ has only distinct, single-multiplicity roots.

Answer 1

The Correct Option is D