If M is the matrix `[(1,-3),(-1,1)]` then find matrix `sum_(r=0)^(oo) ((-1)/3)^(r) M^(r+1)`

Explanation

Let `N=M-M^(2)/3+M^(3)/9-M^(4)/27 +...`
`:. -(MN)/(3) =- M^(2)/3+M^(3)/9-M^(4)/27 ... oo`
Subtracting, we get
`N+(MN)/3=M`
`implies (I+M/3) N=M`
`implies (3I+M)N=3M`
`implies N=[(3,-9),(-3,3)][(4,-3),(-1,4)]^(-1)`
`=1/13 [(3,-9),(-3,3)][(4,-3),(-1,4)]^(-1)`
`=1/13 [(3,-27),(-9,3)]`
`=3/13 [(1,-9),(-3,1)]`