Question

`A=[0 1 3 0]a n dA^8+A^6+A^2+I V=[0 11](w h e r eIi s` the `2xx2` identity matrix`),` then the product of all elements of matrix `V` is _____.

Answer 1

`A=[(0,1),(3,0)]`
`implies A^(2)=A.A=[(0,1),(3,0)][(0,1),(3,0)]=[(3,0),(0,3)]`
`implies A^(4)=A^(2). A^(2)=[(3,0),(0,3)][(3,0),(0,3)]=[(3^(2),0),(0,3^(2))]`
`implies A^(8)=[(3^(4),0),(0,3^(4))]`
and `A^(6)=A^(4). A^(2)=[(3^(2),0),(0,3^(2))][(3,0),(0,3)]=[(3^(3),0),(0,3^(3))]`
Let `V=[(x),(y)]`
`A^(8)+A^(6)+A^(4)+A^(2)+I`
`[(81,0),(0,81)]+[(27,0),(0,27)]+[(9,0),(0,9)]+[(3,0),(0,3)]+[(1,0),(0,1)]`
`=[(121,0),(0,121)]`
`(A^(8)+A^(6)+A^(4)+A^(2)+I)V=[(0),(11)]`
or `[(121,0),(0,121)][(x),(y)]=[(0),(11)]`
or `[(121x),(121y)]=[(0),(11)]`
`implies x=0` and `y=1//11`
`implies V=[(x),(y)]=[(0),(1//11)]`