To understand why a matrix with a non-trivial null space cannot be invertible, we have to look at what the null space actually does to vectors during multiplication.
1. What a Non-Trivial Null Space Means
The null space (or kernel) of a matrix A is the set of all vectors x that get squished into the zero vector (0) when multiplied by A.
A trivial null space contains only the zero vector (x = 0).
A non-trivial null space means there is at least one non-zero vector (v) that gets sent to zero:
A * v = 0 (where v is not equal to 0)
If a matrix A is invertible, it means an inverse matrix, written as A^-1, exists. The fundamental property of an inverse is that it can completely undo the transformation caused by A.
If you multiply a vector by A, you should always be able to multiply it by A^-1 to get your original vector back:
A^-1 * (A * x) = x
Suppose A has a non-trivial null space, so A * v = 0 for some vector v that is not equal to 0.
Now, let's assume A is also invertible, meaning A^-1 exists.
Multiply both sides of our first equation by A^-1:
A^-1 * (A * v) = A^-1 * 0
Look at both sides of this new equation:
Left side: A^-1 * (A * v) simplifies to just v (because the inverse cancels out A).
Right side: A^-1 * 0 must equal 0 (multiplying any matrix by the zero vector results in the zero vector).
Putting them together gives us:
v = 0
We started with the rule that v is not equal to 0 because the null space is non-trivial. We arrived at a contradiction.
