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Nilpotent Matrix
Nilpotent Matrix is a square matrix such that the product of the matrix with itself is equal to a null matrix. A square matrix M of order n × n is termed as a nilpotent matrix if M^{k} = 0. Here k is the exponent of the nilpotent matrix and is lesser than or equal to the order of the matrix( k < n). The order of a nilpotent matrix is n × n, and it easily satisfies the condition of matrix multiplication.
Let us learn more bout the nilpotent matrix, properties of the nilpotent matrix, and also check the examples, FAQs.
1.  What Is A Nilpotent Matrix? 
2.  Properties of Nilpotent Matrix 
3.  Examples on Nilpotent Matrix 
4.  Practice Question 
5.  FAQs on Nilpotent Matrix 
What Is A Nilpotent Matrix?
A nilpotent matrix is a square matrix A such that A^{k} = 0. Here k is called the index or exponent of the matrix, and 0 is a null matrix with the same order as that of matrix A. The matrix multiplication operation is useful to find if the given matrix is a nilpotent matrix or not. Further, the exponent of a nilpotent matrix is lesser than or equal to the order of the matrix (k < n).
The nilpotent matrix is a square matrix with an equal number of rows and columns and it satisfies the condition of matrix multiplication. For a square matrix of order 2 x 2, to be a nilpotent matrix, the square of the matrix should be a null matrix, and for a square matrix of 3 x 3, to be a nilpotent matrix, the square or the cube of the matrix should be a null matrix.
Let us check a few examples, for a better understanding of the working of a nilpotent matrix.
Examples of Nilpotent Matrix
1. An example of 2 × 2 Nilpotent Matrix is A = \(\begin{bmatrix}4&4\\4&4\end{bmatrix}\)
A^{2} = \(\begin{bmatrix}4&4\\4&4\end{bmatrix}\) × \(\begin{bmatrix}4&4\\4&4\end{bmatrix}\)
= \(\begin{bmatrix}4×4+(4)×4&4×(4)+(4)×(4)\\4×4 + (4)× 4&4×(4) + (4)×(4)\end{bmatrix}\)
= \(\begin{bmatrix}16  16&16 + 16\\16  16&16 + 16\end{bmatrix}\)
= \(\begin{bmatrix}0&0\\0&0\end{bmatrix}\)
2. A ndimensional triangular matrix with zeros along the main diagonal can be taken as a nilpotent matrix.
A = \(\begin{bmatrix}0&3&2&1\\0&0&2&2\\0&0&0&3\\0&0&0&0\end{bmatrix}\)
A^{2} = \(\begin{bmatrix}0&0&6&12\\0&0&0&6\\0&0&0&0\\0&0&0&0\end{bmatrix}\)
A^{3} = \(\begin{bmatrix}0&0&0&18\\0&0&0&0\\0&0&0&0\\0&0&0&0\end{bmatrix}\)
A^{4} = \(\begin{bmatrix}0&0&0&0\\0&0&0&0\\0&0&0&0\\0&0&0&0\end{bmatrix}\)
3. Also, a matrix without any zeros can also be referred as a nilpotent matrix. The following is a general form of a nonzero matrix, which is a nilpotent matrix.
A = \(\begin{bmatrix}p&p&p&p\\q&q&q&q\\r&r&r&r\\(p + q + r)&(p + q + r)&(p + q + r)&(p + q + r)\end{bmatrix}\)
Let A = \(\begin{bmatrix}3&3&3\\4&4&4\\7&7&7\end{bmatrix}\)
A^{2} = \(\begin{bmatrix}3&3&3\\4&4&4\\7&7&7\end{bmatrix}\) × \(\begin{bmatrix}3&3&3\\4&4&4\\7&7&7\end{bmatrix}\)
= \(\begin{bmatrix}3×3+3×4+3×(7)&3×3+3×4+3×(7)&3×3+3×4+3×(7)\\4×3+4×4+4×(7)&4×3+4×4+4×(7)&4×3+4×4+4×(7)\\(7)×3+(7)×4+(7)×(7)&(7)×3+(7)×4+(7)×(7)&(7)×3+(7)×4+(7)×(7)\end{bmatrix}\)
= \(\begin{bmatrix}9+1221&9+1221&9+1221\\12 + 16  28&12 + 16  28&12 + 16  28\\21 28 + 49&21 28 + 49&21 28 + 49\end{bmatrix}\)
= \(\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\end{bmatrix}\)
= \(\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\end{bmatrix}\)
Properties of Nilpotent Matrix
Nilpotent matrix is a square matrix, which on multiplying with itself results in a null matrix. The following are some of the important properties of nilpotent matrices.
 The nilpotent matrix is a square matrix of order n × n.
 The index of a nilpotent matrix having an order of n ×n is either n or a value lesser than n.
 All the eigenvalues of a nilpotent matrix are equal to zero.
 The determinant or the trace of a nilpotent matrix is always zero.
 The nilpotent matrix is a scalar matrix.
 The nilpotent matrix is noninvertible.
Related Topics
The following topics help in a better understanding of the nilpotent matrix.
Examples on Nilpotent Matrix

Example 1: Find if the matrix M = \(\begin{bmatrix}2&1\\4&2\end{bmatrix}\) is a nilpotent matrix.
Solution:
The given matrix is M = \(\begin{bmatrix}2&1\\4&2\end{bmatrix}\)
To find if it is a nilpotent matrix, let us square this given matrix.
M^{2} = \(\begin{bmatrix}2&1\\4&2\end{bmatrix}\) × \(\begin{bmatrix}2&1\\4&2\end{bmatrix}\)
= \(\begin{bmatrix}2×2+(1)×4&2×(1)+(1)×(2)\\4×2 + (2)×4&4×(1)+(2)×(2)\end{bmatrix}\)
= \(\begin{bmatrix}4  4&2+2\\88&4+4\end{bmatrix}\)
= \(\begin{bmatrix}0&0\\0&0\end{bmatrix}\)
Therefore, the matrix M is a nilpotent matrix.

Example 2: Compute if the matrix A =\(\begin{bmatrix}0&3&1\\0&0&3\\0&0&0\end{bmatrix}\) is a nilpotent matrix.
Solution:
The given matrix is A = \(\begin{bmatrix}0&3&1\\0&0&3\\0&0&0\end{bmatrix}\)
To find if the matrix is nilpotent we need to find the square and cube of the matrix.
A^{2} = \(\begin{bmatrix}0&0&9\\0&0&0\\0&0&0\end{bmatrix}\)
A^{3} = \(\begin{bmatrix}0&0&9\\0&0&0\\0&0&0\end{bmatrix}\)
Therefore, the matrix A is a nilpotent matrix.
FAQs on Nilpotent Matrix
What Is A Nilpotent Matrix?
A nilpotent matrix is a square matrix A. such that the exponent of A to is a null matrix, and A^{k} = 0. Here k is called the index or exponent of the matrix, and 0 is a null matrix, having the same order as that of matrix A. The matrix multiplication operation is useful to find if the given matrix is a nilpotent matrix or not.
What Is the Nilpotent Matrix Formula?
The formula of a nilpotent matrix for a matrix A is A^{k} = 0. Here the product of the matrix A with itself, for multiple times is equal to a null matrix. Here k is the exponent and for a matrix A of order n × n, the value of k is lesser than or equal to n.
How Do You Find If A Matrix Is A a Nilpotent Matrix?
The given matrix can be tested for it to be a nilpotent matrix or not if the product of the matrix with itself is equal to a null matrix. For a square matrix of order 2, the square of the matrix should be a null matrix, and for a matrix of order 3, the square or the cube of the matrix should be equal to a null matrix.
What Is The Order Of Nilpotent Matrix?
The order of a nilpotent matrix is n x n, and it is a square matrix. For a nilpotent matrix to find the product of the matrix with itself, the given matrix has to be multiplied by itself, and a square matrix with equal number of rows and columns satisfies the condition of matrix multiplication.
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