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Nilpotent Matrices are special types of square matrices, they are special because the product of a Nilpotent Matrix with itself is equal to a null matrix. Let’s take a  square matrix A of order n × n it is considered a nilpotent matrix if Ak = 0. Here k is always less than equal to n.

## Nilpotent Matrix Definition

A square matrix is said to be a nilpotent matrix if the product of the matrix with itself is equal to a null matrix. In simple words, a square matrix “A” of order “n × n” is said to be nilpotent if “Ak = O,” where “O” is a null matrix of order “n × n” and “k” is a positive integer less than or equal to n. A nilpotent matrix is a square matrix that has an equal number of rows and columns and also it satisfies matrix multiplication. For example, if “P” is a nilpotent matrix of order “2 × 2,” then its square must be a null matrix. If “P” is a nilpotent matrix of order “3 × 3,” then either its square or cube must be a null matrix.

### Examples of a Nilpotent Matrix

• Matrix given below is a nilpotent matrix of order “2 × 2.”    • Matrix given below is a nilpotent matrix of order “3 × 3.” As the order of the given matrix is “3 × 3,” then either its square or cube of the matrix must be a null matrix if it is nilpotent. Now, let us find its square first.  Square of the matrix is not a null matrix. So, let us find its cube now. We can see that cube of the matrix “B” is a null matrix. So, the given matrix “B” is nilpotent.

## Properties of a Nilpotent Matrix

Following are some important properties of a nilpotent matrix:

• A nilpotent matrix is always a square matrix of order “n × n.”
• Nilpotency index of a nilpotent matrix of order “n × n” is always equal to either n or less than n.
• Both the trace and the determinant of a nilpotent matrix are always equal to zero.
• As the determinant of a nilpotent matrix is zero, it is not invertible.
• Null matrix is the only diagonalizable nilpotent matrix.
• A nilpotent matrix is a scalar matrix.
• Any triangular matrices with zeros on the principal diagonal are also nilpotent.
• Eigenvalues of a nilpotent matrix are always equal to zero.

Also, Check

## Solved Examples on Nilpotent Matrix

Example 1: Verify whether the matrix given below is nilpotent or not. Solution:

Order of the given matrix is “3 × 3.” If the given matrix is nilpotent, then either its square or cube of the matrix must be a null matrix.

Now, let us find its square first.   Square of the matrix is not a null matrix. So, let us find its cube now.   We can see that cube of the matrix “P” is a null matrix. So, the given matrix “P” is nilpotent.

Example 2: Verify whether the matrix given below is nilpotent or not. Solution:

The order of the given matrix is “2 × 2.” If the given matrix is nilpotent, then its square must be a null matrix.   We can see that square of the matrix “M” is a null matrix. So, the given matrix “M” is nilpotent.

Example 3: Determine whether the matrix given below is nilpotent or not. Solution:

Order of the given matrix is “3 × 3.” If the given matrix is nilpotent, then either its square or cube of the matrix must be a null matrix. Now, let us find its square first.  The square of the matrix is not a null matrix. So, let us find its cube now.   We can see that cube of the matrix “A” is a null matrix. So, the given matrix “A” is nilpotent.

## FAQs on Nilpotent Matrix

### Question 1: What is meant by a nilpotent matrix?

A square matrix is said to be a nilpotent matrix if the product of the matrix with itself is equal to a null matrix.

### Question 2: Is a nilpotent matrix singular?

Yes, a nilpotent matrix is singular, as the determinant of the nilpotent matrix is always zero.