I'm fairly new to linear algebra and I'm trying to make sense of what I'm being taught in class. I'm a little confused as to what happens when all the values of a matrix's row are equal to $0$: does this mean that the matrix has infinite solutions, no solutions, or something completely different?
[Math] What does it mean when all the values of a row in a matrix are 0
linear algebramatrices
Best Answer
Geometrically, if you have an all zero vector in a matrix (row or column; doesn't matter), it means that it represents a dimension-crushing transformation. For instance, a three-dimensional matrix with an all zero row will take three-dimensional vectors and project them into a some two-dimensional plane in three-dimensional space.
The converse isn't true: not all dimension-crushing matrices announce themselves by exhibiting such obvious zeros.
Such transformations are trap door functions: they have no inverse. Therefore, the corresponding matrices are not invertible. Matrices which have no inverse have a determinant of zero (and it's obvious from the way the determinant is calculated that this is so for matrices that are cross-cut by an all-zero row or column). Non-invertible matrices are called singular, or degenerate.
So, if you understand this dimension-crushing aspect, it's easy to see that $Ax = b$ might have no solutions, or many solutions, if $A$ is degenerate. Suppose that $A$ is a 3D matrix that crushes vectors into a 2D plane. Now suppose that vector $b$ lies outside of the plane. Well, then there are no solutions! There is no vector $x$ such that the matrix $A$ will project it onto $b$, because $A$ takes everything into some particular plane, and $b$ is not in that plane. Now suppose that $b$ is inside that plane. Then there are many solutions, because many vectors from 3D space project to any given point on that plane, including $b$.
So all of this gives you a whole new geometric view on systems of linear equations.