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If-x-y-and-z-be-the-pth-qth-and-rth-terms-of-an-AP-show-that-determinant-p-q-r-x-y-z-1-1-1-0-

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Question Number 175838 by Rasheed.Sindhi last updated on 08/Sep/22
If x,y and z be the pth, qth and rth terms of an AP, show that determinant ((p,q,r),(x,y,z),(1,1,1))=0
Ifx,yandzbethepth,qthandrthtermsofanAP,showthat|pqrxyz111|=0
Answered by Rasheed.Sindhi last updated on 08/Sep/22
determinant ((p,q,r),(x,y,z),(1,1,1))=0 determinant ((p,q,r),((a+(p−1)d),(a+(q−1)d),(a+(q−1)d)),(1,1,1)) = determinant ((p,q,r),((a−d+pd),(a−d+qd),(a−d+qd)),(1,1,1)) = determinant ((p,q,r),((a−d),(a−d),(a−d)),(1,1,1))[R2−R1∙d] =(a−d) determinant ((p,q,r),(1,1,1),(1,1,1))=(a−d)∙0=0 [Two rows are identical] Proved
|pqrxyz111|=0|pqra+(p−1)da+(q−1)da+(q−1)d111|=|pqra−d+pda−d+qda−d+qd111|=|pqra−da−da−d111|[R2−R1⋅d]=(a−d)|pqr111111|=(a−d)⋅0=0[Tworowsareidentical]Proved
Answered by som(math1967) last updated on 08/Sep/22
determinant ((p,q,r),((a+pd−d),(a+qd−d),(a+rd−d)),(1,1,1)) C_1 ^l →C_1 −C_2 ,C_2 ^l →C_2 −C_3 determinant (((p−q),(q−r),r),(((p−q)d),((q−r)d),(a+rd−d)),(0,0,1)) (p−q)(q−r) determinant ((1,1,r),(d,d,(a+rd−d)),(0,0,1)) =(p−q)(q−r)×0 [C_1 ,C_2 identical] =0
|pqra+pd−da+qd−da+rd−d111|C1l→C1−C2,C2l→C2−C3|p−qq−rr(p−q)d(q−r)da+rd−d001|(p−q)(q−r)|11rdda+rd−d001|=(p−q)(q−r)×0[C1,C2identical]=0
Commented by Rasheed.Sindhi last updated on 08/Sep/22
Good sir, thank you!
Goodsir,thankyou!

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