Nilpotent matrix calculator. It will also find the determinant, inverse, rref (reduced row echelon form), null space, rank, eigenvalues, and eigenvectors and will multiply the matrix by a scalar. A square matrix A such that A^n is the zero matrix 0 for some positive integer matrix power n, known as the index (Ayres 1962, p. Let us learn more about the properties and examples of the nilpotent matrix. Tool to calculate the Jordan Normal Form of a Matrix (by Jordan reduction of a square matrix) to get, by decomposition, 2 matrices S and J such that M = S . Aug 21, 2025 · As the order of the given matrix is "3 × 3," then either its square or cube of the matrix must be a null matrix if it is nilpotent. The calculator will find the null space (kernel) and the nullity of the given matrix, with steps shown. 3. In this case, we say that the matrix columns (or rows) are collinear. Just type matrix elements and click the button. 2. Find a nilpotent $2\times 2$ matrix of index 2. A square matrix `A` is called a nilpotent matrix, if `A^m = 0` for some positive integer m. Explanation of what a nilpotent matrix is, with all its properties and examples of nilpotent matrices (2x2, 3x3,). All nilpotent elements are zero divisors. Feb 14, 2026 · There are two equivalent definitions for a nilpotent matrix. Instead of just giving a final answer, Symbolab’s AI Math Solver breaks problems down step-by-step. Formula to find a nilpotent matrix. The square matrix A is said to be nilpotent if A n = A ⁢ A ⁢ ⋯ ⁢ A ⏟ n times = 𝟎 for some positive integer n (here 𝟎 denotes the matrix where every entry is 0). If is nilpotent, then is a unit, because entails More generally, the sum of a unit element and a nilpotent element is a unit when they commute. You can use decimal fractions or mathematical expressions: Nilpotent matrix by Marco Taboga, PhD A square matrix is said to be nilpotent if, by rasing it to a sufficiently high integer power, we get the zero matrix as a result. Supports matrices up to 10×10. A matrix A is said to be a nilpotent matrix if A^k = 0. A square matrix whose eigenvalues are all 0. All About Symbolab’s Online AI Math Solver & Calculator Getting the right answer in math is important, but what really matters is knowing how you got there, and being able to do it again the next time. Another equivalent definition is that a matrix is singular when at least one column (or row) of the matrix is a linear combination of other columns (or rows) of that matrix. Leave extra cells empty to enter non-square matrices. Now, let us find its square first. Feb 21, 2026 · Matrix Trace Calculator - Calculate the trace of a square matrix (sum of diagonal elements), verify its equality with the sum of eigenvalues, explore trace properties, and visualize the diagonal with an interactive heatmap. A matrix is singular if and only if its determinantis zero. [5] This calculator will add, subtract, multiply, divide, and raise to power two matrices, with steps shown. With help of this calculator you can: find the matrix determinant, the rank, raise the matrix to a power, find the sum and the multiplication of matrices, calculate the inverse matrix. That’s what Symbolab’s AI Math Solver is built for. Nilpotent matrix is a square matrix such that the product of the matrix with itself is equal to a null matrix. Share this solution or page with your friends. An matrix with entries from a field is nilpotent if and only if its characteristic polynomial is . 1. Matrix Calculator: A beautiful, free matrix calculator from Desmos. No nilpotent element can be a unit (except in the trivial ring, which has only a single element 0 = 1). 11). Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels. It shows what to do first, how . com. A singular matrix is also called to be degener These matrices are nilpotent but there are no zero entries in any powers of them less than the index. J . S̄ Oct 29, 2025 · Learn about Nilpotent Matrix with clear definitions, step-by-step explanations, and examples to help you understand and master this concept easily. Show that it is nilpotent of the correct index. ozb uai qis bjs pbo leb tbf ewe fsu ebu qqq dzf pzm jwq dhs