Show that A =[{:(5,3),(-1,-2):}] satisfi...

Show that `A =[{:(5,3),(-1,-2):}]` satisfies the equation `A^(2)-3A-7I=0` and hence find the value of `A^(-1)`

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We have. `A=[{:(5,3),(-1,-2):}]`
`therefore A^(2)=A.A=[{:(5,3) ,(-1,-1) :}][{:(5,3),(-1,-2):}]`
`=[{:(25-3,15-6),(-5+2,-3+4):}]=[{:(22,9),(-3,1):}]`
`3A=3[{:(5,3),(-1,-2):}]=[{:(15,9),(-3,-6):}]`
and `7I=7[{:(22,9),(-3,1):}]-[{:(15,9),(-3,-6):}]-[{:(7,0),(0,7):}]`
`=[{:(22-15 -7,9-9-0),( -3 +3-0,1+ 6-7):}]`
` =[{:(0,0) ,(0,0):}]`
=0
Since, `A^(2)-3A-7I=0`
`rArr A^(-1)[A(A^(2))-3A-7I]=A^(-1)0`
`rArr A^(-1),A-3A^(-1)-7A^(-1)I=0`
`rArr IA-3I-7A^(-1)=0 [because A^(-1)A=I]`
`rArr A-3I-7A^(-1)=0 [because A^(-1)I=A^(-1)]`
`rArr -7A^(-1)=-A+3I`
`= [{:(-5,-3),(1,2):}]+[{:(3,0),(0,3):]=[{:(-2,-3),(1,5):}]`
`therefore A^(-1)=(-1)/(7)[{:(-2,-3),(1,5):}]`

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