Sarrus Rule Calculator guide
How to Check Whether a 3×3 Matrix Is Invertible
A 3×3 matrix is invertible exactly when its determinant is not zero. Compute the determinant — with the rule of Sarrus this takes six multiplications — and if the result is anything other than zero, the inverse exists. If it is zero, the matrix is singular and has no inverse.
The test is one number
This is one of the tidiest results in linear algebra, so it is worth stating plainly: compute the determinant, and compare it to zero. Non-zero means invertible. Zero means singular. There is nothing else to check.
You do not need to try building the inverse to see whether it works. You do not need the rank, or the eigenvalues, or anything else. One determinant settles it.
The matrix loaded into the calculator above has a determinant of −1, so it is invertible.
Everything that single number tells you
A non-zero determinant is equivalent to a whole list of other statements. They are all true together, or all false together, so establishing any one of them establishes the rest. If a question asks you to show any of these, computing the determinant is often the shortest honest route:
- The matrix has an inverse.
- The rows are linearly independent, and so are the columns.
- The matrix has full rank — which is 3, for a 3×3.
- The system Ax = b has exactly one solution for every possible b.
- Zero is not an eigenvalue of the matrix.
- As a transformation, it does not flatten any dimension away.
Why the determinant is what decides it
The formula for a matrix inverse divides by the determinant. When that number is zero the formula breaks down, and it is not that we lack a better formula — no inverse exists to be found.
The picture version is the one that tends to stick. A zero determinant means the transformation flattens 3D space down onto a plane or lower. Information about where points came from is destroyed in the process, and no transformation can put it back. That is exactly what having no inverse means.
One caution about very small determinants
A determinant of 0.0000001 is not zero, so strictly the matrix is invertible. But if your numbers came from measurements rather than from a textbook, that distinction can be misleading.
Matrices with determinants very close to zero are called ill-conditioned. They technically have inverses, but tiny errors in the input produce enormous errors in the output, so the inverse you compute may be worthless in practice.
For hand calculations with whole numbers this never comes up, and you can safely set it aside. It starts to matter the moment your matrix comes from real-world data.
