# Consider the system of equations

Consider the following system of equations

$$x_1+2x_2 =k$$

$$3x_1+4x_2+x_3 =1$$

$$5x_1+6x_2+2x_3=2$$

where $$k$$ is an undetermined constant. For what values of $$k$$, if any, is the system inconsistent?

Our goal is to construct an augmented matrix for these equations, simplify by performing some row operations, and to see if we can infer something about $$k$$. $$\left[\begin{array}{rrr|r} 1 & 2 & 0 & k \\ 3 & 4 & 1 & 1 \\ 5 & 6 & 2 & 2 \end{array}\right]$$

As Tanner suggested, multiply the second row by 2 ($$2R_2$$):

$$\left[\begin{array}{rrr|r} 1 & 2 & 0 & k \\ 6 & 8 & 2 & 2 \\ 5 & 6 & 2 & 2 \end{array}\right]$$

We now see that the last two columns of Row 2 reduce to 0 if we subtract Row 3 from it (i.e. $$R_2-R_3$$):

$$\left[\begin{array}{rrr|r} 1 & 2 & 0 & k \\ 1 & 2 & 0 & 0 \\ 5 & 6 & 2 & 2 \end{array}\right]$$

And now the coefficient sides of $$R_1$$ and $$R_2$$ are identical, so we subtract $$R_1-R_2$$ to get $$k$$ on its own:

$$\left[\begin{array}{rrr|r} 0 & 0 & 0 & k \\ 1 & 2 & 0 & 0 \\ 5 & 6 & 2 & 2 \end{array}\right]$$

We now see that if $$k\neq0$$, there are no possible solutions (note that if $$k=0$$ then we would have infinitely many solutions). The system is therefore inconsistent if $$k\neq0$$.

Hint:

Multiply the second equation by $$2$$ and subtract the third equation.

What must $$k$$ be to be consistent?

In the eq. $$AX=B$$ if $$|A|=0$$ but at least one of the Cramer's determinants $$|A_1|, | ,A_2|,|A_3|$$ is non-zero. $$|A|=\left |\begin{array}{ccc} 1 & 2 & 0 \\ 3 & 4 & 1 \\ 5 & 6 & 2\end{array} \right |=0$$ $$|A_1|=\left |\begin{array}{ccc} k & 2 & 0 \\ 2 & 4 & 1 \\ 2 & 6 & 2\end{array} \right |=2k-4$$ $$|A_2|= \left| \begin{array}{ccc} 1 & k & 0\\ 3 & 2 & 1 \\5 & 2 & 2 \end{array} \right|=2-k$$ $$\left| \begin{array}{ccc} 1 & 2 & k \\3 & 4 & 2 \\ 5 & 6 & 2 \end{array}\right|=4-2k$$

So If $$k=2$$ there are many solutions and for $$k \ne 2$$ there will be no solution and the system will be inconsistent.