If we have the following matrix:
$
\begin{bmatrix}
a & b & c\\
0 & e-\frac{bd}{a} & f-\frac{cd}{a}\\
0 & h-\frac{bg}{a} & i-\frac{cg}{a}
\end{bmatrix}
$
Why, given that the first entry in the first column is non-zero, and the other two entries are 0s, is it that the columns of the matrix are linearly dependant if the columns in the minor of a:
$
\begin{bmatrix}
e-\frac{bd}{a} & f-\frac{cd}{a}\\
h-\frac{bg}{a} & i-\frac{cg}{a}
\end{bmatrix}
$
are linearly dependent?
Best Answer
First of all, the number of linearly independent columns is the same as the number of linearly independent rows, which is in itself not a trivial thing.
Once you know that, you can observe that the first row cannot be written as a linear combination of the second and third rows (since it has a non-zero component in the first entry, whereas the other two have zeros in their first component). This means that the first row is independent of the second and third ones.
Now you need to check if the second and third rows are linearly independent. Since the first entries are zeros, they do not count and you are left with the rows of the matrix you have written.
Since, again, the number of linearly independent rows and columns is the same, you're done.