Question

If `"A" = [(a , b),(c , d)] and "I" = [(1 , 0),(0 , 1)]` show that A² - (a + d) A = (bc - ad) I.

Answer 1

Here A² - (a + d) A
= `[(a , b),(c , d)][(a , b),(c , d)] -(a + d)[(a , b),(c ,d)]`

= `[(a^2 + bc , ab + bd),(ac + dc, cb + d^2)] - [(a^2 + ad , ab + bd),(ac + dc , ad + d^2)]`

= `[(bc - ad , 0),(0 , bc -ad)] = (bc - ad)[(1 , 0),(0 , 1)]`

= (bc - ad) I.
Hence proved.