यदि `A={:[(1, 2,3),(3,-2,1),(4,2,1)]:}` है, तो दर्शाइए कि `A^(3)-23A-40I=0`.

Explanation

`A^(2)=AxxA={:[(1,2,3),(3,-2,1),(4,2,1)][(1,2,3),(3,-2,1),(4,2,1)]:}`
`={:[(1+6+12,2-4+6,3+2+30),(3-6+4,6+4+2,9-2+1),(4+6+4,8-4+2,12+2+1)]=[(19,4,8),(1,12,8),(14,6,15)]:}`
`A^(3)=A^(2)xxA=[(19,4,8),(1,12,8),(14,6,15)][(1,2,3),(3,-2,1),(4,2,1)]:}`
`={:[(19+12+32,38-8+16,57+4+8),(1+36+32,2-24+16,3+12+8),(14+18+60,28-12+30,42+6+15)]=[(63,46,49),(69,-6,23),(92,46,63)]:}`
`A^(3)-23A-40I={:[(63,46,69),(69,-6,23),(92,46,63)]-23[(1,2,3),(3,-2,1),(4,2,1)]-40[(1,0,0),(0,1,0),(0,0,1)]:}`
`={:[(63,46,69),(69,-6,23),(92,46,63)]-[(23,46,69),(69,-46,23),(92,46,23)]-[(40,0,0),(0,40,0),(0,0, 40)]:}`
`={:[(63-23-40,46-46+0,69-69-0),(69-69-0,-6+46-40,23-23-0),(92-92-0,46-46-0,63-23-40)]:}`
`={:[(0,0,0),(0,0,0),(0,0,0)]=0`.