show that `[(1,1,3),(5,2,6),(-2,-1,-3)]=A` is nipotent matrix of order

1.	show that `[(1,1,3),(5,2,6),(-2,-1,-3)]=A` is nipotent matrix of order 3.
Answer» Let `A=[(1,1,3),(5,2,6),(-2,-1,-3)]` `therefore A^(2)=A.A=[(1,1,3),(5,2,6),(-2,-1,-3)]xx[(1,1,3),(5,2,6),(-2,-1,-3)]` `[(1+5-6,1+2-3,3+6-9),(5+10-12,5+4-6,15+12-18),(-2-5+6,-2-2+3,-6-6+9)]` `=[(0,0,0),(3,3,9),(-1,-1,-3)]` `thereforeA^(3)=A^(2).=[(0,0,0),(3,3,9),(-1,-1,-3)]xx[(1,1,3),(5,2,6),(-2,-1,-3)]` `[(0+0+0,0+0+0,0+0+0),(3+15-18,3+6-9,9+18-27),(-1-5+16,-1-2+3,-3-6+9)]=[(0,0,0),(0,0,0),(0,0,0)]=0` `therefore" " A^(3)=Oi.e.,A^(k)=O` Here `k=3`

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show that `[(1,1,3),(5,2,6),(-2,-1,-3)]=A` is nipotent matrix of order 3.

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