1. The problem is to understand what a nilpotent matrix is and what is meant by the period of a nilpotent matrix. 2. A nilpotent matrix $N$ is a square matrix such that for some positive integer $k$, the matrix raised to the power $k$ is the zero matrix: $$N^k = 0.$$ This means that multiplying the matrix by itself $k$ times results in the zero matrix. 3. The smallest such positive integer $k$ for which this holds is called the period or the index of nilpotency of the matrix. 4. For example, if $N^3 = 0$ but $N^2 \neq 0$, then the period of $N$ is 3. 5. The period tells us how many times we need to multiply the nilpotent matrix by itself before it becomes the zero matrix. 6. This concept is important in linear algebra and appears in the study of Jordan normal forms and matrix exponentials. Final answer: A nilpotent matrix is a matrix $N$ such that $N^k = 0$ for some smallest positive integer $k$, and this $k$ is called the period (or index) of the nilpotent matrix.