Question

A square matrix A is said to be a nilpotent matrix of degree $r$, if $r$ is the least positive integer such that $A^r=0$. 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

The correct option is B $AB=BA$

Given : $A$ is a nilpotent matrix of degree $r,$ such that $A^r=0.$

$\Rightarrow$ Now, $A\xi B$ are nilpotent matrices, then $(A+B)$ will be nilpotent matrix if $AB=BA.$

$\Rightarrow$ Now, $A$ is a nilpotent matrix and $B$ is also nilpotent matrix so that the product is also nilpotent. And according to the law of multiplicity $AB=BA.$

Hence, the answer is $AB=BA.$