24thx sir..but in book soln was a bit long...
first they found A'(x),,then some row and column transfrmations to get A'(x)=ΣΣAlk
and then integrated to get A(x)=xΣΣAlk +c
putting x=0,=>c=A(0)
=> A(x)=xΣΣAlk +A(0)
Is this also OK ?
Q1 A_x=\begin{bmatrix} a_{11}+x & a_{12}+x & a_{13}+x\\ a_{21}+x & a_{22}+x &a_{23}+x \\ a_{31}+x & a_{32}+x & a_{33}+x \end{bmatrix}
Prove A_x=A(0)+x\sum_{k=1}^{3}{\sum_{l=1}^{3}{A_{lk}}}
Q2 If a,b>0 and a3+b3=a-b,then prove a2+b2<1
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2 Answers
62observation: The given determinant is a linear equation in x
so it is of the form ax+b
to find b, simply put x=0
we get that b=A(0) as needed.
Now, to get the value of a,
put x=1 you will get the desired result
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