Matrix Properties Checker
Nilpotent, idempotent, involutory, orthogonal, symmetric, skew-symmetric — tested by actually multiplying the matrix out, with A² and the decisive power printed as the proof. Trace and determinant alongside.
The default 3×3 is the shift matrix: A sends (x, y, z) to (0, x, y), A² to (0, 0, x), and A³ to zero — a classic nilpotent.
How we got it — the example, step by step
- A sends (x, y, z) to (0, x, y): everything falls onto the yz plane. A² sends it to (0, 0, x) — a single 1 survives, in the bottom-left corner. A³ = 0: everything reaches the origin.
- The tool computes A² through A⁶ entry by entry and tests each property against those actual products — the matrices printed above are the proof, not an illustration.
Check it backwards
Questions people also ask
Can a matrix have several of these properties at once?
Yes — the identity is idempotent, involutory, orthogonal and symmetric simultaneously. But nilpotent plus idempotent forces the zero matrix: A = A² = A⁴ = … = Aᵏ = 0.
Is the zero matrix nilpotent?
Yes, trivially — A¹ = 0 already, index 1. The page labels it as the trivial case instead of dressing it up.
Why does the check stop at A⁶?
Margin. Cayley–Hamilton guarantees a nilpotent 3×3 is dead by A³; checking to A⁶ costs nothing and removes any doubt about rounding at the boundary — entries are compared to zero within 10⁻⁹.