If A `=[{:(0,1,0),(0,0,1),(p,q,r):}]`, show that ltbargt `A^(3)= pI+qA+rA^(2)`

Explanation

we have, `A(2)=A.A`
`[(0,1,0),(0,0,1),(p,q,r)]xx[(0,1,0),(0,0,1),(p,q,r)]`
`[(0,0,1),(p,q,r),(pr,p+qr,q+r^(2))]`
`therefore (3)=A^(2)=[(0,1,1),(p,q,r),(pr,p+qr,q+r^(2))]xx[(0,1,0),(0,0,1),(p,q,r)]`
`=[(p,q,r),(pr,p+qr,q+r^(2)),(pq+r^(2)p,pr+q^(2)+qr^(2),p+2qr+r^(3))]" "therefore(i)`
and `pI+qA+rA^(2) =p[(1,0,0),(0,1,0),(0,0,1)]+q[(0,1,0),(0,0,1),(p,q,r)]+r[(0,0,1),(p,q,r),(pr,p+qr,q+r^(2))]`
`[(p+0+0,0+q+0,0+0+r),(0+0+pr,p+0+qr,0+q+r^(2)),(0+pq+pr^(2),0+q^(2)+pr+qr^(2),p+2qr+r^(3))]`
`[(p,q,r),(pr,p+qr,q+r^(3)),(pq+pr^(2),q^(2)+pr+qr^(2),p+2qr+r^(3))]" "therefore(ii)`
thus, from Eqs. (i) and (ii), we get `A = pI+qA+rA^(2)`