If `a+(1)/(b)=1` and `b+(1)/(c)=1` then show that `c+1/a=1`

If 1 + (1)/(a) + (1)/(b) + (1)/(c) = 0 , then Delta = |(1 +a,1,1),(1,1 +b,1),(1,1,1 +c)| is equal to

If (1)/(a)+(1)/(c )=(1)/(2b-a)+(1)/(2b-c) , then

If (b+c)/a,(c+a)/b,(a+b)/c are in A.P. show that 1/a,1/b1/c are also in A.P. (a +b+c != 0) .

If (b+c)^(-1), (c+a)^(-1), (a+b)^(-1) are in A.P. then show that (a)/(b+c) , (b)/(c+a) , (c)/(a+b) are also in A.P.

If ax^2 + 2bx + c = 0 and x^2 + 2b_(1)x + c_(1) = 0 have a common root and a/a_(1), b/b_(1),c/c_(1) are in show that a_(1), b_(1), c_(1) are in G.P.

if A=[(1,2),(3,4):}],B=[{:(-1,0),(2,3):}]and C=[{:(1,-1),(0,1):}], then show that : (i) A(B+C)=AB+AC (ii) (A-B)C=AC-BC.

Find the inverse of the matrix A=[(a, b),( c,(1+b c)/a)] and show that a A^(-1)=(a^2+b c+1)I-a A .

(i) if A=[{:(1,0),(0,1):}],B=[{:(0,1),(1,0):}]and C=[{:(1,0),(0,1):}], then show that A^(2)=B^(2)=C^(2)=I_(2). (ii) if A=[{:(1,0),(1,1):}],B=[{:(2,0),(1,1):}]and C=[{:(-1,2),(3,1):}], then show that A(B+C)=AB+AC. (iii) if A=[{:(1,-1),(-1,1):}]and B=[{:(1,1),(1,1):}], then show that AB is a zero matrix.

If a is the A.M. between b and c, b the G.M. between a and c, then show that 1/a,1/c,1/b are in A.P.

If a ,b and c are all non-zero and |(1+a,1,1),( 1,1+b,1),(1,1,1+c)|=0, then prove that 1/a+1/b+1/c+1=0