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Let A=[[a,b],[c,d]] be a 2xx2 real matri...

Let `A=[[a,b],[c,d]]` be a `2xx2` real matrix. If `A-alphaI` is invertible for every real number `alpha` , then

Text Solution

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`A-alpha*I=[[a,b],[c,d]]-[[alpha,0],[0,alpha]]`
`=[[a-alpha,b],[c,d-alpha]]`
`=(a-alpha)(d-alpha)-bc!=0`
`alpha^2-alpha(a+d)+ad-bc`
`(a+d)^2-4(ad-bc)<0`
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