Question

If `A=[-1 1 0-2]` , then prove that `A^2+3A+2I=Odot` Hence, find `Ba n dC` matrices of order 2 with integer elements, if `A=B^3+C^3dot`

Answer 1

`A=[(-1,1),(0,-2)]`
`implies A^(2)=[(-1,1),(0,-2)][(-1,1),(0,-2)]=[(1,-3),(0,4)]`
`implies A^(2)+3A+2I`
`=[(1,-3),(0,4)]+3[(-1,1),(0,-2)]+2[(1,0),(0,1)]=[(0,0),(0,0)]`
`implies A^(2)+3A+2I =O` (1)
From (1), `A^(3)+3A^(2)+2A=O`
`implies (A+I)^(3)-A=I^(3)`
`implies A=(A+I)^(3)-I^(3)=(A+I)^(3)+(-I)^(3)`
`implies B=A+I` and `C=-I`
`:. B=[(-1,1),(0,-2)]+[(1,0),(0,1)]=[(0,1),(0,-1)]`
and `C=[(-1,0),(0,-1)]`