Computing an inverse and using it

Invertible: all the ways to say it

The word "invertible" can be said in ten different ways. They all mean the same thing — and being able to move between them is what makes linear algebra efficient.

The equivalence theorem

For a square matrix A of size n × n, the following statements are all equivalent:

1.  A is invertible
2.  det(A) ≠ 0
3.  rank(A) = n
4.  the columns of A are linearly independent
5.  the columns of A form a basis of R^n
6.  A X = 0 has only the zero solution
7.  A X = B has a unique solution, for every B
8.  elimination yields n pivots (no zero row)
9.  0 is not an eigenvalue of A
10. the associated linear map is bijective

Each is checked by a different computation, but they stand or fall together. In practice, pick whichever is fastest to test in the situation at hand:

small matrix (2×2, 3×3)      ->  the determinant
arbitrary matrix             ->  elimination: count the pivots
visibly dependent columns    ->  conclude immediately (not invertible)
already diagonalised matrix  ->  check whether 0 is an eigenvalue

The geometric thread

All these formulations describe the same phenomenon from different angles:

              A NOT invertible
                     |
    ______________________________________
   |            |             |            |
 det = 0     rank < n     dependent      kernel
   |            |          columns      not just 0
   |            |             |            |
    \__________ space is CRUSHED _________/
                     |
            and a collapse cannot be undone

Conversely, an invertible matrix preserves dimension: nothing is lost, everything is reached, the transformation is reversible.

A useful special case: triangular matrices

[ 2   7  -1 ]
[ 0   3   5 ]      invertible  <=>  no zero on the diagonal
[ 0   0   4 ]

Since the determinant is the product of the diagonal, the answer can be read off with no computation at all. This is very handy, because elimination produces precisely triangular matrices.

In practice: do not compute the inverse

Here is a point theory courses often skip. To solve A X = B numerically, one almost never computes A⁻¹:

Computing A⁻¹ then A⁻¹B   ->  about 3 times more operations,
                              and it amplifies rounding errors
Running Gaussian elimination -> fewer operations, more stable

The inverse remains an essential theoretical tool — it gives the formula X = A⁻¹B, it characterises well-posed systems, it appears in proofs. But the solving algorithm is still elimination. That is what numpy.linalg.solve(A, B) does, always preferable to inv(A) @ B.

One borderline case deserves caution: a matrix can be invertible in theory and catastrophic in practice, if its determinant is tiny. It is then called ill-conditioned: a minute change in B completely changes the solution. The determinant says whether you are invertible; it does not say how far you are from not being so.

Summary