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A square matrix A is said to be a nilpotent matrix of degree r

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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 + B = A B

A B = B A

A - B = A B

none of these

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Elijah Discussed

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

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Text solutionVerified

Given :

A

is a nilpotent matrix of degree

r,

such that

A^{r} = 0.

\Rightarrow

Now,

A ξ B

are nilpotent matrices, then

(A + B)

will be nilpotent matrix if

A B = B A .

\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

A B = B A .

Hence, the answer is

A B = B A .

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Question Text	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
Answer Type	Text solution:1
Upvotes	150

A square matrix A is said to be a nilpotent matrix of degree r, if r is the least positive integer such that Ar=0. If A and B are nilpotent matrices then A+B will be a nilpotent matrix if

Text solutionVerified

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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