Question

Show that there do not exist scalars c1, c2, and c3 such that c1(1, 0, 1, 0) + c2(1, 0, -2, 1) + c3(2, 0, 1, 2) = (1, -2, 2, 3)

Answer 1

Write the system in augmented-matrix form:

$c_1(1,0,1,0)+c_2(1,0,-2,1)+c_3(2,0,1,2)=(1,-2,2,3)$

$\iff\left[\begin{array}{ccc|c}1&1&2&1\\0&0&0&-2\\1&-2&1&2\\0&1&2&3\end{array}\right]$

Row reduce this matrix:

• Add -1(row 1) to row 3:

$\left[\begin{array}{ccc|c}1&1&2&1\\0&0&0&-2\\0&-3&-1&1\\0&1&2&3\end{array}\right]$

• Add 3(row 4) to row 3:

$\left[\begin{array}{ccc|c}1&1&2&1\\0&0&0&-2\\0&0&5&10\\0&1&2&3\end{array}\right]$

• Multiply row 3 by 1/5:

$\left[\begin{array}{ccc|c}1&1&2&1\\0&0&0&-2\\0&0&1&2\\0&1&2&3\end{array}\right]$

• Add -2(row 3) to row 4:

$\left[\begin{array}{ccc|c}1&1&2&1\\0&0&0&-2\\0&0&1&2\\0&1&0&-1\end{array}\right]$

• Add -2(row 3) and -1(row 4) to row 1:

$\left[\begin{array}{ccc|c}1&0&0&-2\\0&0&0&-2\\0&0&1&2\\0&1&0&-1\end{array}\right]$

This matrix tells us that $c_1=-2$, $c_2=-1$, and $c_3=2$, but clearly $0c_1+0c_2+0c_3=0\neq-2$, so there is no solution.