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Nilpotent Matrix

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Match the following {:(,"Column-I",,"Column-II",),((A),"If A and B are otthogonal, then AB is",(p),"Zero matrix",),((B),"If A and B are nilpotent matrices of order r and s and A and B commute, then "(AB)^(r)"is",(q),"Nilpotent matrix",),((C),"If A is a hermitian matrix such that "A^(2)=0", then A is ",(r),"Unitary matrix",),((D),"If A and B are unitary matrices, then AB is",(s),"Orthogonal",):}

(A) If A and B are orthogonal,then AB is (B) If A and B are nilpotent matrices of order r and s and A and B commute,then (AB)^(r) is

Knowledge Check

  • Some special square matrices are defined as follows. Nilpotent matrix: A square matrix. A is said to be rilpotent (of order 2)it, A^(2)=O . A squre matrix is said to be nilpotent of order p, if p is the least positive integer such that A^(p)=O . Idempotent martrix: A square matrix A is said tto be idempotent it, A^(2)=A . e.g.[{:(,1,0),(,0,1):}] is an idempotent matrix. Involutory matrix: A square A is said to be involutnary if A^(2)=I , I being the identify matrix. e.g..A=[{:(,1,0),(,0,1):}] is an involutary matrix. Orothogonal matrix: A square matrix A is said to be an orthogonal matrix it A' A=I=A A' The matrix A [{:(,1,1,3),(,5,2,6),(,-2,-1,-3):}]

    A
    idempotent matrices
    B
    involutary matrices
    C
    nilpotent matrix
    D
    none of these

  • Some special square matrices are defined as follows. Nilpotent matrix: A square matrix. A is said to be rilpotent (of order 2)it, A^(2)=O . A squre matrix is said to be nilpotent of order p, if p is the least positive integer such that A^(p)=O . Idempotent martrix: A square matrix A is said tto be idempotent it, A^(2)=A . e.g.[{:(,1,0),(,0,1):}] is an idempotent matrix. Involutory matrix: A square A is said to be involutnary if A^(2)=I , I being the identify matrix. e.g..A=[{:(,1,0),(,0,1):}] is an involutary matrix. Orothogonal matrix: A square matrix A is said to be an orthogonal matrix it A' A=I=A A' If A and B are two square matrices such that AB=A&BA then A&B are

    A
    idempotent matrices
    B
    involutary matrices
    C
    Orthogonal matrices
    D
    Nilpotent matrices

  • Some special square matrices are defined as follows. Nilpotent matrix: A square matrix. A is said to be rilpotent (of order 2)it, A^(2)=O . A squre matrix is said to be nilpotent of order p, if p is the least positive integer such that A^(p)=O . Idempotent martrix: A square matrix A is said tto be idempotent it, A^(2)=A . e.g.[{:(,1,0),(,0,1):}] is an idempotent matrix. Involutory matrix: A square A is said to be involutnary if A^(2)=I , I being the identify matrix. e.g..A=[{:(,1,0),(,0,1):}] is an involutary matrix. Orothogonal matrix: A square matrix A is said to be an orthogonal matrix it A' A=I=A A' If [{:(,0,2beta,gamma),(,alpha,beta,-gamma),(,alpha,-beta,gamma):}] is orthogonal, then

    A
    `alpha=pm(1)/(sqrt2)`
    B
    `beta=pm(1)/(sqrt6)`
    C
    `gamma=pm(1)/(sqrt3)`
    D
    all of those

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    A square matrix A is said to be nilpotent of induare if A^(m)=0. Now if for this A,(I-A)^(n)=I+A+A^(2)+A^(3)++A^(m-1) then n is equal to

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