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.
In this article, we will learn about the Nilpotent Matrix in detail.
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?
Answer:
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?
Answer:
Yes, a nilpotent matrix is singular, as the determinant of the nilpotent matrix is always zero.
Question 3: What is the order of a Nilpotent Matrix?
Answer:
The order of a nilpotent matrix is determined by the number of rows and columns that it has. In general, a nilpotent matrix has an equal number of rows and columns because it is a square matrix.
Question 4: How to find whether a matrix is nilpotent or not?
Answer:
To find whether the given matrix is nilpotent or not, we have to check if the product of the matrix with itself is a null matrix. 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.