**Determine the conditions for which the system x+y+z=1; x+2y-z=b; 5x+7y+az=b²;
Has (i)Unique solution (ii)infinite number of solutions (iii)no solution**
Now going into the way I solved Let A be the coefficient matrix...
Now as per Cramer's rule for unique solution of the system only the det(A)=0 but now if one looks into the matrix theory... Let A' as shown be the corresponding augmented matrix then...
The system will have unique solution when rank A=rank A'=no. of variables in the system...
Now following the matrix theory if we solve for the unique solution case then apart from a not equals to 1 ( which actually is the answer) we get another condition that is b is not equals to either 3 or -1...
Why is this discrepancy arising??
From your notes, it looks like you calculated $A'$ only for the degenerate case $a=1$, which you know from Cramer's rule to not have a unique solution (I see no variable $a$ in those calculations anywhere).
In that case the special values for $b$ distinguish between 'no solution' and 'infinitely many solutions'.
What you get is that for $a=1$ and $b=3,-1$
rank $A =$ rank $A' = 2$, but since the number of variables is $3$, this does not contradict your matrix theory characterization of unique solutions.