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Because the product of a Nilpotent Matrix and itself equals a null matrix, Nilpotent Matrix is a unique variety of square matrix. Let A be a square matrix of order *n* × *n*. If A^{k} = 0, A is a nilpotent matrix. In this case, *k* is never equal to *n*.

We shall study the Nilpotent Matrix in detail in this post.

**The Definition of Nilpotent Matrix**

Nilpotent matrix is a square matrix which will produce a null matrix if we multiply it with itself.

A square matrix is a nilpotent matrix if A^{k} = 0, where A is a square matrix with a dimension (order) of *n* x *n*, and *k* is the exponent. The value of *k* is less than or equal to the dimension or order of *n*.

Thus,

A nilpotent matrix is a square matrix with *n* x *n* that fulfills A^{k} = 0, where *k* ≤ *n*.

It is a bit different with an idempotent matrix.

**Example of Nilpotent Matrix**

In order to make us understand better, let us put it into practice.

Assume matrix A is a square matrix with the order of 2

**Properties of Nilpotent Matrix**

- An order n × n square matrix is the nilpotent matrix.
- A nilpotent matrix with an order of
*n*×*n*has an index that is either*n*or less than*n*. - A nilpotent matrix has eigenvalues that are all equal to zero.
- A nilpotent matrix’s trace, or determinant, is always zero.
- A scalar matrix is the nilpotent matrix.
- There is no way to invert the nilpotent matrix.

**Nilpotent Matrix Index Calculation**

The smallest value of “k” is known as the index of the Nilpotent matrix. The definition states that if a square matrix [A] is a Nilpotent matrix, it will satisfy the equation A^{k} = 0 for certain positive values of “k.”

Therefore, all you have to do is multiply matrix [A] by the same matrix again until you obtain a zero matrix, or null matrix (0), in order to find the index of the nilpotent matrix. Assume, for instance, that after multiplying matrix [A] by k times, you obtained A^{k} = 0.

As a result, that integer value k will be the index of that Nilpotent matrix [A].

The *n* x *n* Nilpotent matrix’s index will always be at most n times its value. Therefore, you will need to multiply the maximal matrix by *n* (the matrix’s order).