How can we determine if the hyper-plane pass through the origin? Let $A$ is an  $n \times n$ matrix.  Consider each row of $A$ as a point in $\mathbb{R}^n$; and assume these $n$ points define a unique hyperplane in $\mathbb{R}^n$ that passes through each point.
The question is how to determine or what is the condition to find whether the hyperplane pass through the origin?

As user has written $det A=0$ is the condition. It may be convenient for you to think in terms of 3 dimension first. To define a plane in 3 dimension we require 3 points to be specified, let the points be $(x_1,x_2,x_3)$. Now the equation of the plane can be expressed as $A_1x+A_2y+A_3z+A=0$ where $A$ is the determinant of the points $(x_1,x_2,x_3)$ and $A_i$ is the determinant obtained by replacing $x_i$ with a column vector of $1$. 
Now you should look at the general equation of the plane expressed as $A_1x+A_2y+A_3z+A=0$, if the origin is in the plane then the determinant of $A$ is must be zero otherwise equation will not be satisfied. If this is extended to $n$ dimension you should get your answer. Hope it helps.
