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

Explanation

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`