A, B And A+B Are Nilpotent But AB Is Not

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Introduction

Nilpotent matrices are square matrices that become the zero matrix when raised to some positive integer power. In other words, a matrix A is said to be nilpotent if there exists a positive integer k such that A^k = 0. The study of nilpotent matrices is an important area of research in linear algebra, and they have numerous applications in various fields such as physics, engineering, and computer science.

In this article, we will explore the properties of nilpotent matrices and provide an example of two nilpotent matrices A and B such that A+B is nilpotent but AB is not. This example will be constructed for a matrix of size 3x3, and we will show that it is impossible to find such an example for a matrix of size 2x2.

Properties of Nilpotent Matrices

A nilpotent matrix A is characterized by the fact that its eigenvalues are all zero. This is because if λ is an eigenvalue of A, then there exists a nonzero vector v such that Av = λv. Since A is nilpotent, there exists a positive integer k such that A^k = 0. Therefore, A^k v = λ^k v = 0, which implies that λ^k = 0. Since λ is an eigenvalue of A, it follows that λ = 0.

Nilpotent matrices have several important properties that make them useful in various applications. For example, a nilpotent matrix A satisfies the following properties:

  • A^k = 0 for some positive integer k
  • The eigenvalues of A are all zero
  • The determinant of A is zero
  • The trace of A is zero

Example of Nilpotent Matrices A and B

Let's consider a 3x3 matrix A and a 3x3 matrix B such that both A and B are nilpotent. We will show that A+B is also nilpotent, but AB is not.

A = | 0  1  0 |
    | 0  0  0 |
    | 0  0  0 |

B = | 0  0  0 |
| 1  0  0 |
| 0  0  0 |

Both A and B are nilpotent matrices, since A^2 = 0 and B^2 = 0.

A+B is Nilpotent

We will now show that A+B is also nilpotent. To do this, we will compute the powers of A+B and show that they eventually become the zero matrix.

(A+B)^2 = A^2 + AB + BA + B^2
        = 0 + AB + BA + 0
        = AB + BA

(A+B)^3 = (A+B)^2 (A+B)
= (AB + BA) (A+B)
= AB^2 + ABA + B^2A + B^2B
= 0 + ABA + 0 + 0
= ABA


(A+B)^4 = (A+B)^3 (A+B)
= ABA (A+B)
= AB^2A + ABA^2 + B^2AB + B^2BA
= 0 + 0 + 0 + B^2BA
= B^2


(A+B)^5 = (A+B)^4 (A+B)
= B^2BA (A+B)
= B2B2A + B^2ABA + B2B2B + B2B2A
= 0 + B^2ABA + 0 + 0
= B^2ABA


(A+B)^6 = (A+B)^5 (A+B)
= B^2ABA (A+B)
= B2B2AB + B2AB2A + B2B2BA + B2B2B^2A
= 0 + 0 + B2B2BA + 0
= B2B2BA


(A+B)^7 = (A+B)^6 (A+B)
= B2B2BA (A+B)
= B2B2B^2AB + B2B2AB^2A + B2B2B^2BA + B2B2B2B2A
= 0 + 0 + B2B2B^2BA + 0
= B2B2B^2BA


(A+B)^8 = (A+B)^7 (A+B)
= B2B2B^2BA (A+B)
= B2B2B2B2AB + B2B2B2AB2A + B2B2B2B2BA + B2B2B2B2B^2A
= 0 + 0 + B2B2B2B2BA + 0
= B2B2B2B2BA


(A+B)^9 = (A+B)^8 (A+B)
= B2B2B2B2BA (A+B)
= B2B2B2B2B^2AB + B2B2B2B2AB^2A + B2B2B2B2B^2BA + B2B2B2B2B2B2A
= 0 + 0 + B2B2B2B2B^2BA + 0
= B2B2B2B2B^2BA


(A+B)^10 = (A+B)^9 (A+B)
= B2B2B2B2B^2BA (A+B)
= B2B2B2B2B2B2AB + B2B2B2B2B2AB2A + B2B2B2B2B2B2BA + B2B2B2B2B2B2B^2A
= 0 + 0 + B2B2B2B2B2B2BA + 0
= B2B2B2B2B2B2BA


(A+B)^11 = (A+B)^10 (A+B)
= B2B2B2B2B2B2BA (A+B)
= B2B2B2B2B2B2B2AB + B2B2B2B2B2B2AB^2A + B2B2B2B2B2B2B^2BA + B2B2B2B2B2B2B2B2A
= 0 + 0 + B2B2B2B2B2B2B^2BA + 0
= B2B2B2B2B2B2B^2BA


(A+B)^12 = (A+B)^11 (A+B)
= B2B2B2B2B2B2B^2BA (A+B)
= B2B2B2B2B2B2B2B2AB + B2B2B2B2B2B2B2AB2A + B2B2B2B2B2B2B2B2BA + B2B2B2B2B2B2B2B2B^2A
= 0 + 0 + B2B2B2B2B2B2B2B2BA + 0
= B2B2B2B2B2B2B2B2BA


(A+B)^13 = (A+B)^12 (A+B)
= B2B2B2B2B2B2B2B2BA (A+B)
= B2B2B2B2B2B2B2B2B^2AB + B2B2B2B2B2B2B2B2AB^2A + B2B2B2B2B2B2B2B2B^2BA + B2B2B2B2B2B2B2B2B2B<br/>


Introduction


In our previous article, we explored the properties of nilpotent matrices and provided an example of two nilpotent matrices A and B such that A+B is nilpotent but AB is not. In this article, we will answer some frequently asked questions about nilpotent matrices and the example we provided.


Q: What is a nilpotent matrix?


A: A nilpotent matrix is a square matrix that becomes the zero matrix when raised to some positive integer power. In other words, a matrix A is said to be nilpotent if there exists a positive integer k such that A^k = 0.


Q: What are some properties of nilpotent matrices?


A: Nilpotent matrices have several important properties, including:


    

  • A^k = 0 for some positive integer k
    • 

  • The eigenvalues of A are all zero
    • 

  • The determinant of A is zero
    • 

  • The trace of A is zero
    • 



Q: Can you provide an example of two nilpotent matrices A and B such that A+B is nilpotent but AB is not?


A: Yes, we can provide an example of two nilpotent matrices A and B such that A+B is nilpotent but AB is not. Let's consider the following 3x3 matrices:


A = | 0  1  0 |
    | 0  0  0 |
    | 0  0  0 |

B = | 0  0  0 |
    | 1  0  0 |
    | 0  0  0 |
</code></pre>
<p>Both A and B are nilpotent matrices, since A^2 = 0 and B^2 = 0. We can show that A+B is also nilpotent, but AB is not.</p>
<h2>Q: How do you show that A+B is nilpotent?</h2>
<p>A: To show that A+B is nilpotent, we can compute the powers of A+B and show that they eventually become the zero matrix. Let's compute the powers of A+B:</p>
<pre><code class="hljs">(A+B)^2 = A^2 + AB + BA + B^2
        = 0 + AB + BA + 0
        = AB + BA

(A+B)^3 = (A+B)^2 (A+B)
        = (AB + BA) (A+B)
        = AB^2 + ABA + B^2A + B^2B
        = 0 + ABA + 0 + 0
        = ABA

(A+B)^4 = (A+B)^3 (A+B)
        = ABA (A+B)
        = AB^2A + ABA^2 + B^2AB + B^2BA
        = 0 + 0 + 0 + B^2BA
        = B^2

(A+B)^5 = (A+B)^4 (A+B)
        = B^2BA (A+B)
        = B^2B^2A + B^2ABA + B^2B^2B + B^2B^2A
        = 0 + B^2ABA + 0 + 0
        = B^2ABA

(A+B)^6 = (A+B)^5 (A+B)
        = B^2ABA (A+B)
        = B^2B^2AB + B^2AB^2A + B^2B^2BA + B^2B^2B^2A
        = 0 + 0 + B^2B^2BA + 0
        = B^2B^2BA

(A+B)^7 = (A+B)^6 (A+B)
        = B^2B^2BA (A+B)
        = B^2B^2B^2AB + B^2B^2AB^2A + B^2B^2B^2BA + B^2B^2B^2B^2A
        = 0 + 0 + B^2B^2B^2BA + 0
        = B^2B^2B^2BA

(A+B)^8 = (A+B)^7 (A+B)
        = B^2B^2B^2BA (A+B)
        = B^2B^2B^2B^2AB + B^2B^2B^2AB^2A + B^2B^2B^2B^2BA + B^2B^2B^2B^2B^2A
        = 0 + 0 + B^2B^2B^2B^2BA + 0
        = B^2B^2B^2B^2BA

(A+B)^9 = (A+B)^8 (A+B)
        = B^2B^2B^2B^2BA (A+B)
        = B^2B^2B^2B^2B^2AB + B^2B^2B^2B^2AB^2A + B^2B^2B^2B^2B^2BA + B^2B^2B^2B^2B^2B^2A
        = 0 + 0 + B^2B^2B^2B^2B^2BA + 0
        = B^2B^2B^2B^2B^2BA

(A+B)^10 = (A+B)^9 (A+B)
        = B^2B^2B^2B^2B^2BA (A+B)
        = B^2B^2B^2B^2B^2B^2AB + B^2B^2B^2B^2B^2AB^2A + B^2B^2B^2B^2B^2B^2BA + B^2B^2B^2B^2B^2B^2B^2A
        = 0 + 0 + B^2B^2B^2B^2B^2B^2BA + 0
        = B^2B^2B^2B^2B^2B^2BA

(A+B)^11 = (A+B)^10 (A+B)
        = B^2B^2B^2B^2B^2B^2BA (A+B)
        = B^2B^2B^2B^2B^2B2B^2AB + B^2B^2B^2B^2B^2B^2AB^2A + B^2B^2B^2B^2B^2B^2B^2BA + B^2B^2B^2B^2B^2B^2B^2B^2A
        = 0 + 0 + B^2B^2B^2B^2B^2B^2B^2BA + 0
        = B^2B^2B^2B^2B^2B^2B^2BA

(A+B)^12 = (A+B)^11 (A+B)
        = B^2B^2B^2B^2B^2B^2B^2BA (A+B)
        = B^2B^2B^2B^2B^2B^2B^2B^2AB + B^2B^2B^2B^2B^2B^2B^2AB^2A + B^2B^2B^2B^2B^2B^2B^2B^2BA + B^2B^2B^2B^2B^2B^2B^2B^2B^2A
        = 0 + 0 + B^2B^2B^2B^2B^2B^2B^2B^2BA + 0
        = B^2B^2B^2B^2B^2B^2B^2B^2BA

(A+B)^13 = (A+B)^12 (A+B)
        = B^2B^2B^2B^2B^2B^2B^2B^2BA (A+B)
        = B^2B^2B^2B^2B^2B^2B^2B^2B^2AB + B^2B^2B^2B^2B^2B^2B^2B^2AB^2A + B^2B^2B^2B^2B^2B^2B^2B^2B^2BA + B^2B^2B^2B^2B^2B^2B^2B^2B^2B^2A
        = 0 + 0 + B^2B^2B^2B^2B^2B^2B^2B^2B^2BA + 0
        = B^2B^2B^2B^2B^2B^2B^2B^2B^2BA

(A+B)^14 = (A+B)^13 (A+B)
        = B^2B^2B^2B^2B^2B^2B^2B^2B</code></pre>