Nilpotent Matrix Checker

Check whether repeatedly multiplying a square matrix by itself eventually produces the all-zero matrix.

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Summary

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Detail

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Check whether repeatedly multiplying a square matrix by itself eventually produces the all-zero matrix.

How It Works

How Nilpotent Matrix Checker Works

The calculator repeatedly multiplies the matrix by itself — A, A², A³, and so on — checking after each step whether the result has become the all-zero matrix. By a result called the Cayley-Hamilton theorem, an n×n nilpotent matrix is guaranteed to reach zero by at most the n-th power, so the calculator stops looking beyond that point.

Worked Example

See It In Action

The matrix with rows 0,1 / 0,0 squares to the all-zero matrix, so it's nilpotent with index 2.

Real-World Use Cases

Who Uses Nilpotent Matrix Checker and Why

Checking whether a strictly triangular matrix (zeros on and above/below the diagonal) actually reduces to zero under repeated multiplication, as expected.
Verifying a matrix used in a Jordan form or eigenvalue exercise is nilpotent as part of confirming its eigenvalues are all zero.
Confirming a matrix constructed for a linear algebra proof or homework problem genuinely satisfies the nilpotent definition rather than just looking sparse.

Common Mistakes

Mistakes to Avoid

Assuming a matrix with several non-zero entries can't be nilpotent — plenty of non-zero entries are allowed; what matters is that some power of the whole matrix eventually becomes exactly zero.
Expecting an invertible matrix to ever qualify — a nilpotent matrix's determinant is always zero (since some power of it equals the zero matrix, whose determinant is zero), so nilpotent and invertible are mutually exclusive.
Not realizing the check needs to test successive powers (A, A², A³, ...) rather than just A itself — a matrix can look nowhere near zero at first power and still be nilpotent.

Pro Tips

Tips for Best Results

By the Cayley-Hamilton theorem, an n×n nilpotent matrix is guaranteed to reach the zero matrix by at most the n-th power, so there's a hard upper bound on how far the check needs to look.
Strictly triangular matrices (zero on the diagonal too, not just above or below it) are always nilpotent — a fast mental check if you already know a matrix has that shape.

Troubleshooting

Fixing Common Problems

My matrix never reaches zero no matter how many powers I check. — If it hasn't reached zero by the n-th power for an n×n matrix, it isn't nilpotent — the Cayley-Hamilton theorem guarantees any genuinely nilpotent matrix reaches zero within that many steps.

Glossary

Terms Explained

Nilpotency index: The smallest power k for which a nilpotent matrix Aᵏ equals the zero matrix.

Cayley-Hamilton theorem: A result guaranteeing that an n×n nilpotent matrix reaches the zero matrix by at most its n-th power, giving a natural stopping point for the check.

FAQ

Frequently Asked Questions

Can a nilpotent matrix have a non-zero entry?

Yes — plenty of individual entries can be non-zero; what matters is that the matrix eventually cancels itself out entirely once multiplied by itself enough times.

Can an invertible matrix ever be nilpotent?

No — a nilpotent matrix always has determinant zero (since some power of it is the zero matrix, whose determinant is zero), so it can never be invertible.