Periodic matrix and Nilpotent matrix and involutory matrix - Practice Questions & MCQ

Periodic matrix
If a square matrix A satisfies the relation $\mathrm{A}^{\mathrm{k}+1}=\mathrm{A}$, where k is a +ve integer. Then A is called a periodic matrix. If k is the least + ve integer for which this condition is satisfied then k is called the period of A. For $\mathrm{k}=1$, we get $\mathrm{A}^2=\mathrm{A}$, which is the condition for idempotent matrix, so the period of idempotent matrix $=1$.

Nilpotent matrix
If $A$ satisfies the condition $A^k=O$ and $A^{k-1} \neq O$, then $A$ is called nilpotent matrix and $k$ is known as the order of nilpotent matrix $A$.

Involutory matrix
If A satisfies the condition $\mathrm{A}^2=1$, where I is the identity matrix then A is called the involutory matrix.
Note: $A=A^{-1}$ for involutory matrix.