# Nilpotent Matrix Calculator Online

### Explanation:

Square matrices that are nilpotent have a product that is equal to a empty or null matrix when the matrix is multiplied by itself. A nilpotent matrix is a square matrix (M) of order nxn if it satisfies Mk = 0. The nilpotent matrix's exponent, k, is less than or equal to the matrix's order (k < n) in this case. A nilpotent matrix has an order of nxn and readily meets the requirement of matrix multiplication.

### What is An Nilpotent Matrix?

A square matrix A is a nilpotent matrix if Ak = 0. In this case, k is referred to as the matrix's index or exponent, and 0 is a null or empty matrix with the same order as of matrix A. To determine whether or not the given matrix is a nilpotent matrix or not, the matrix multiplication procedure is helpful. Additionally, the order of a nilpotent matrix's exponent (k < n) is less than or equal to the exponent of the matrix.
The nilpotent matrix is you can say is a square matrix that meets the requirement for matrix multiplication and has an equal number of rows and columns. In order for a square matrix of order 2 x 2 to be a nilpotent matrix, the square of the given matrix must be a null matrix, and for a square matrix of order 3 x 3, the cube must be a null matrix.

### Properties of Nilpotent Matrix

A square matrix known as a nilpotent matrix produces a null matrix when multiplied by itself. Some of the key characteristics of nilpotent matrices are the ones listed below.

• The nilpotent matrix must be a square matrix having order n x n.
• A nilpotent matrix with an order of nxn has an index that can only be lesser than or equal to n.
• A nilpotent matrix has zero for each of its eigenvalues.
• A nilpotent matrix's determinant or trace is always zero.
• A scalar matrix makes up the nilpotent matrix.
• The nilpotent matrix cannot be reversed or you can say it is non-invertible.

#### How to use check Nilpotent Matrix Calculator?

• Firstly, you need to enter the dimension of the matrix. Enter number of rows in "Rows" input field and Enter number of columns in "Columns" input field.
• Then press the button "Set Matrix".
• An empty matrix will appear below and then you can enter your values inside the matrix.
• After entering all the values press "Solve" button, the result will automatically appear below which check whether the matrix is Nilpotent matrix or not.

Nilpotent Matrix Example Image:

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