Please answer 21, 22 and 23 ques - Maths - Matrices

Question

Please answer 21, 22 and 23 ques

Rishabh Mittal · Accepted Answer

Dear Student ,

Please find below the solution to the asked query :

22

Given A= αβ01Let pn : An=αnβ αn-1α-101For n=1A1=α1β α1-1α-101A= αβ01So, p1 is true 
Let us assume that pn is also true for n=k 
Ak=αkβ αk-1α-101Now ,for n=k+1Ak+1=Ak A=αkβ αk-1α-101  αβ01=αk+1αkβ+β αk-1α-101=αk+1βαk+ αk-1α-101=αk+1βαk+1-αk+αk-1α-101=αk+1βαk+1-1α-101=RHSThus , pk+1 is true
Hence by PMI , pn is true for all n∈ N
Hence Proved

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