TITLE: Properties of a matrix whose row vectors are dependent
QUESTION [8 upvotes]: When a column vector in a matrix is a made up of "combination" of its other column vectors, it is said to be linearly dependant. Say...
$$
A=\begin{bmatrix}
2 & 1 & 0\\ 
4 & 5 & -6\\ 
3 & 1 & 1
\end{bmatrix}
$$
$$
1\begin{bmatrix}
2\\ 
4\\ 
3
\end{bmatrix}-2\begin{bmatrix}
1\\ 
5\\ 
1
\end{bmatrix}=\begin{bmatrix}
0\\ 
-6\\ 
1
\end{bmatrix}

$$
Otherwise, it is linearly independent. And being linearly dependent, it has the properties of being a singular matrix and therefore may have an infinite solutions or no solutions at all depending on the result matrix. Then being linearly independent, the matrix is more often a good matrix that can span the entire $R^{n}$ space and has a unique solution to its system of equations.
Then I just thinking what happens if a row vector in a matrix is made up of "combination" of its other row vectors? Say...
$$P=\begin{bmatrix}
2 & 5 & 1\\ 
12 & 13 & 3\\ 
8 & 3 & 1
\end{bmatrix}
$$
$$
2\begin{bmatrix}
2 & 5 & 1
\end{bmatrix}+1\begin{bmatrix}
8 & 3 & 1
\end{bmatrix}=\begin{bmatrix}
12 & 13 & 3
\end{bmatrix}
$$
Does a matrix have any special properties too if its row vectors are linearly independent and linearly dependent?
Thanks!

REPLY [6 votes]: Yes. The following are equivalent for a square matrix $A$:

$A$ is non-singular
the rows of $A$ are linearly independent
the columns of $A$ are linearly independent