Why every $3\times 3$ matrix with entries $a_{ij} = ci + dj$ is singular? The matrix I got is: 
$$\begin{bmatrix}c+d & c+2d & c+3d \\2c+d & 2c+2d & 2c+3d \\
3c+d & 3c+2d & 3c+3d\\\end{bmatrix}$$
 A: Such a matrix is a sum of two matrices of rank at most one. For example,
$$\pmatrix{c+d & c+2d & c+3d\\
    2c+d & 2c+2d & 2c+3d\\
    3c+d & 3c+2d & 3c+3d}=\pmatrix{c& c & c\\
    2c& 2c & 2c\\
    3c& 3c& 3c}+\pmatrix{d & 2d & 3d\\
    d & 2d & 3d\\
    d & 2d & 3d}.$$
Therefore, its rank is at most two.
A: Using row operations $R'_i = R_i -R_1$, $R''_3 = R'_3 -2R'_2$:
$$
\begin{bmatrix}
    c+d & c+2d & c+3d \\
    2c+d & 2c+2d & 2c+3d \\
    3c+d & 3c+2d & 3c+3d \\
    \end{bmatrix}
\to
\begin{bmatrix}
    c+d & c+2d & c+3d \\
    c & c & c \\
    2c & 2c & 2c \\
    \end{bmatrix}
\to
\begin{bmatrix}
    c+d & c+2d & c+3d \\
    c & c & c \\
    0 & 0 & 0 \\
    \end{bmatrix}
$$
A: The $i$ and $j$ are subscripts.  I rewrote as
$$
W =
\left(
\begin{array}{ccc}
 m+p& m+q  & m+r  \\
 n+p & n+q  & n+r  \\
  o+p & o+q  & o+r  \\
\end{array}
\right)
$$
Now, if $p=q=r$ there is a separate argument. If not all the same, we get
$$
v =
\left(
\begin{array}{c}
 q-r  \\
 r-p \\
  p-q  \\
\end{array}
\right)
$$
is a (nonzero) null vector, $Wv = 0$
