If `A=|{:(1,2,2),(2,1,2),(2,2,1):}|`, then show that `A^(2)-4A-5I_(3)=0`. Hemce find `A^(-1)`.

If `A=|{:(1,2,2),(2,1,2),(2,2,1):}|`, then show that `A^(2)-4A-5I_(3)=

1.	If `A=\|{:(1,2,2),(2,1,2),(2,2,1):}\|`, then show that `A^(2)-4A-5I_(3)=0`. Hemce find `A^(-1)`.
Answer» We leave it to the reader to show that `A^(2)-4A-5I=O`. `Now, A^(2)-4A-5I=O` `rArr" "A A-4A=5I` `rArr" "(A A).A^(-1)-4A.A^(-1)=5I.A^(-1)` `rArr" "A(A A^(-1))-4I=5A^(-1)` `rArr" "AI-4I=5A^(-1)` `A-4I=5A^(-1)` `rArr" "A^(-1)=1/5(A-4I)`. `:." "A^(-1)=1/5.{[(1,2,2),(2,1,2),(2,2,1)]-4.[(1,0,0),(0,1,0),(0,0,1)]}` `=1/5.{[(1,2,2),(2,1,2),(2,2,1)]-[(4,0,0),(0,4,0),(0,0,4)]}` `=1/5.[(-3,2,2),(2,-3,2),(2,2,-3)]`.

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