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Let A = [[0, alpha],[0,0]] and(A+I)^(70)...

Let `A = [[0, alpha],[0,0]] and(A+I)^(70) - 70 A = [[a-1,b-1],[c-1,d-1]],` the
value of ` a + b + c + d ` is

Text Solution

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The correct Answer is:
6
`because A = [[0 ,alpha],[0,0]]`
`therefore A^(2) = Acdot A = [[0 ,alpha],[0,0]][[0 ,alpha],[0,0]]=[[0 ,0],[0,0]]=0`
`rArr A^(2) = A^(3) = A^(4) = A^(5) = ...= 0`
Now, `(A + I) ^(70) = (I+A)^(70)`
`= I + ""^(70)C_(1) A + ""^(70)C_(2) A^(2) + ""^(70)C_(3) A^(3) +...+ ""^(70)C _(70)A^(70`
`= I + 70 A + 0 + 0 + ...=I+70A`
`rArr (A+I) ^(70) - 70 A = I = [[1,0],[0,1]]= [[a-1,b-1],[c-1, d-1]]` [given ]
`therefore a- 1 = 1, b-1 = 0, c-1 = 0, d-1=1`
`rArr a=2, b=1, c=, d=2`
Hence, `a + b + c + d = 6`
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