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Consider the system of equations:
Find all values of c such that the system has:
(a) no solutions
(b) a unique solution
(c) infinitely many solutions
In case c. write the general solution in the parametric vector form.
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Step 1: Write the system of equations in matrix form
In this case
Step 2: Write A and b as augmented matrix.
Augmented form :
Step 3: Reduce the augmented matrix in row echelon form.
After performing the row operations row2 = row2 - row1 and row3 = row3 - row1, we get the following reduced row echelon matrix
For part (a), we can see that in order for the system of equations to have no solution, while and thus for there is no solution. With this value of our augment matrix looks like
indiacting no solution
For part (b), that is, for the system of equations to have unique solution, we do not want any row of our matrix to be zero entries. Thus we want and Thus for all values of except and , we have unique solution.
For part (c), For the system of equation to have an infinitely many solutions, we want at least one free variable, i.e we want at least one row to be all zero entries. This can be achieved if and . Thus with , we have the third row as all zero entries.
Thus our augmented matrix for is
We know that the first column represents entries of , the second column represents entries of and the third column represents entries of .
Thus looking at our augmented matrix, it can be said column 1 and 2 are pivots columns and so only is a free variable.
We solve for general solution as follows: