Analyze matrices through exact arithmetic and dependency testing. See powers, coefficients, degree, and verification instantly. Built for students, proofs, homework, and symbolic exploration daily.
Enter integers, decimals, or fractions such as 3/4. This calculator works with square rational matrices up to 4 × 4.
| Example | Matrix | Expected Minimal Polynomial | Why It Happens |
|---|---|---|---|
| Jordan block | [[2, 1], [0, 2]] | (x - 2)^2 = x^2 - 4x + 4 | A repeated eigenvalue with a nondiagonal Jordan block needs degree 2. |
| Diagonal matrix | [[1, 0], [0, 3]] | (x - 1)(x - 3) = x^2 - 4x + 3 | Distinct diagonal entries require both linear factors. |
| Identity matrix | [[1, 0], [0, 1]] | x - 1 | The identity matrix is annihilated by one linear factor. |
Definition: The minimal polynomial of a square matrix A is the unique monic polynomial m(x) of smallest degree such that m(A) = 0.
Core relation: The calculator searches for the first exact dependence among I, A, A², …, Aᵏ. When it finds
Aᵏ + ck-1Ak-1 + … + c1A + c0I = 0,
it returns
m(x) = xᵏ + ck-1xk-1 + … + c1x + c0.
Characteristic check: The page also computes the characteristic polynomial using the Faddeev–LeVerrier recursion, then verifies that the minimal polynomial divides it.
Exact arithmetic: Fractions are reduced symbolically throughout elimination, multiplication, and verification, so rational inputs stay exact.
It accepts square matrices up to 4 × 4 with rational-style entries. You can type integers, decimals, or fractions such as 7/3.
Exact arithmetic avoids rounding errors that often hide the true annihilating relation. This is especially important for repeated eigenvalues and Jordan blocks.
No. The minimal polynomial always divides the characteristic polynomial, but it can have lower degree. Identity and diagonal matrices often show this difference clearly.
It shows the first exact dependence among I, A, A², and higher powers. That relation directly gives the monic minimal polynomial coefficients.
They let you inspect the algebra behind the answer. You can verify the first dependency step instead of treating the result as a black box.
Yes, provided the decimals represent exact rational values you intend. For example, 0.25 becomes 1/4 automatically before any symbolic elimination begins.
It shows m(A). A correct result should produce the zero matrix. That confirmation is printed beside the polynomial summary for convenience.
That can happen when no lower-degree dependence exists among I, A, A², and later powers. Certain companion or cyclic matrices behave this way.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.