Question

A square matrix A is said to be a nilpotent matrix of degree r if r is the least positive integer such that ${{\text{A}}^{r}}=0$. If A and B are nilpotent matrices, then A + B will be a nilpotent matrix if:
(a) A + B = AB
(b) AB = BA
(c) A – B = AB
(d) None of these

Answer 1

Hint: First define a nilpotent matrix. Next use the property that the product of two nilpotent matrices will also result in a nilpotent matrix. A and B are nilpotent matrices. So, using the above property, AB and BA are also nilpotent matrices. Next use the law of multiplicity to get AB = BA which is our final answer.

Complete step by step answer:
In this question, we are given that a square matrix A is said to be a nilpotent matrix of degree r if r is the least positive integer such that ${{\text{A}}^{r}}=0$.
If A and B are nilpotent matrices, then we have to find the condition such that A + B will be a nilpotent matrix.
We know that the product of two nilpotent matrices will also result in a nilpotent matrix.
Since, A and B are nilpotent matrices. So, using the above property, AB and BA are also nilpotent matrices.
Now, A is a nilpotent matrix and B is also a nilpotent matrix so that the product is also nilpotent. And according to the law of multiplicity AB = BA.
Hence, the final answer is AB = BA.
So, option (b) is correct.

Note: In this question, it is very important to know what the properties of a nilpotent matrix are. Also, note that it is important for a nilpotent matrix to have ${{\text{A}}^{r}}=0$ but at the same time ${{\text{A}}^{r-1}}\ne 0$.