What is the geometric meaning of singular matrix Could anyone help explain what is the geometric meaning of singular matrix ? What's the difference between singular and non-singular matrix ? I know the definition, but couldn't understand it very well.

Thanks.
If you are in $\Bbb{R}^3$, say you have a matrix like

$$\left[\begin{array}{ccc} a_{11} &amp; a_{12} &amp; a_{13} \\ a_{21} &amp; a_{22} &amp; a_{23} \\a_{31} &amp; a_{32} &amp; a_{33} \end{array}\right].$$

Now you can think of the columns of this matrix to be the "vectors" corresponding to the sides of a parallelepiped. If this matrix is singular i.e. has determinant zero, then this corresponds to the parallelepiped being completely squashed, a line or just a point.
In my opinion, zero determinant of a matrix with nonzero entries shows the balance like the resultant of two equal forces in opposite direction is zero or equilibrium state.