If a matrix such that A^2-4A-5I=0, find the value of the constant a&b such that the inverse of aI+bA is A^2
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If a matrix such that A^2-4A-5I=0, find the value of the constant a&b such that the inverse of aI+bA is A^2

[From: ] [author: ] [Date: 11-11-21] [Hit: ]
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A^2-4A-5I = 0 ==> A^2=5I+ 4A

A^2 is the inverse of aI+bA ==> A^2(aI+bA) = I
==> (5I+4A)(aI+bA)=I
==>4bA^2 + (5b+4a)A + (5a-1)I=0
==> 4b(5I+4A) + (5b+4a)A + (5a-1)I=0
==> (21b+4a)A + (5a+20b-1)I = 0

==> 21b+4a=0 and 5a+20b-1=0
==> a= 21/25 and b=-4/25
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