Question

If A = `[(2,-2,2),(2,3,0),(9,1,5)]` then, show that (adj A) A = O.

Answer 1

Given A = `[(2,-2,2),(2,3,0),(9,1,5)]`

[A_ij] = `|(|(3,0),(1,5)|,-|(2,0),(9,5)|,|(2,3),(9,1)|),(-|(-2,2),(1,5)|,|(2,2),(9,5)|,-|(2,-2),(9,1)|),(|(-2,2),(3,0)|,-|(2,2),(2,0)|,|(2,-2),(2,3)|)|`

`= [(15-0,-(10-0),(2-27)),(-(-10-2),(10-18),-(2+18)),(0-6, -(0-4),(6+4))]`

`= [(15,-10,-25),(12,-8,-20),(-6,4,10)]`

adj A = [A_ij]^T = `[(15,12,-6),(-10,-8,4),(-25,-20,10)]`

Now (adj A) A = `[(15,12,-6),(-10,-8,4),(-25,-20,10)] [(2,-2,2),(2,3,0),(9,1,5)]`

`= [(30+24-54, -30+36-6, 30+0-30),(-20-16+36, 20-24+4, -20+0+20),(-50-40+90,50-60+10,-50+0+50)]`

`= [(0,0,0),(0,0,0),(0,0,0)]` = O