Polynomial Matrix Calculator

Calculator input

Large screens use 3 columns, medium screens use 2, and phones use 1.

Matrix A

Enter one row per line. Separate polynomial entries with commas.

Matrix B

Needed for binary operations and all B-specific actions.

Operation Evaluation point x

Graph x minimum

Graph x maximum

Graph points

Example data table

Example item	Matrix or value	Meaning
Matrix A	`x^2+1, 2x-3 -x+4, 3`	A 2×2 polynomial matrix with mixed quadratic and linear entries.
Matrix B	`2x, 1 x^2-1, x+2`	A compatible 2×2 matrix for addition, subtraction, and multiplication.
Evaluation point	`x = 2`	Converts symbolic entries into numeric values for direct inspection.
Expected det(A)	`5x^2 - 11x + 15`	Illustrates a scalar polynomial generated from a matrix determinant.

Formula used

Polynomial addition and subtraction: combine like powers term by term. For example, (2x^2 + x) + (3x^2 - 4) becomes 5x^2 + x - 4.

Polynomial multiplication: multiply each term in the first polynomial by each term in the second, then collect like powers. This is used inside every matrix multiplication step.

Matrix addition and subtraction: C[i,j] = A[i,j] ± B[i,j]. Each corresponding entry remains a polynomial.

Matrix multiplication: C[i,j] = Σ A[i,k]B[k,j]. Each sum contains polynomial products, so the result is still polynomial.

Trace: tr(A) = Σ A[i,i]. The diagonal polynomials are added into one scalar polynomial.

Determinant for 2×2: det(A) = a11a22 - a12a21.

Determinant for 3×3: first-row cofactor expansion is used, so each cofactor is a polynomial minor determinant.

Adjugate: build the cofactor matrix and then transpose it. This is useful when analyzing inverse-related structure, even if the actual inverse is rational.

Evaluation: substitute the selected x into every entry. This creates a numeric matrix, enabling determinant, trace, and norm summaries.

How to use this calculator

Enter Matrix A with one row per line and commas between entries.
Enter Matrix B if the selected action uses both matrices or targets B.
Choose an operation such as addition, multiplication, determinant, trace, adjugate, transpose, or evaluation.
Set the x value for numeric substitution and define the graph interval.
Press the calculate button. The result appears under the header and above the form.
Review the symbolic or numeric output, summary lines, and Plotly graph.
Use the CSV button for table-friendly export and the PDF button for a shareable snapshot.

FAQs

1) What matrix sizes are supported?

General addition, subtraction, multiplication, transpose, and evaluation work for many dimensions. Determinant and adjugate are implemented for square matrices up to 3×3.

2) How should I type each polynomial?

Use forms like x^2+1, -3x, 4, or 0.5x^3-2x+7. Spaces are allowed, and uppercase X is converted automatically.

3) Why is the result symbolic instead of numeric?

Most operations keep polynomial structure intact. Choose an evaluation option when you want the matrix converted at a specific x value.

4) What does the graph represent?

Scalar results plot directly against x. Matrix results plot summary measures, including trace, determinant when available, and Frobenius norm.

5) Can I compute an inverse matrix here?

Not directly. Polynomial matrix inverses often contain rational expressions. The adjugate and determinant still help you analyze invertibility and structure.

6) Why do I get a dimension error?

Addition and subtraction need equal dimensions. Multiplication needs the number of columns in A to match the number of rows in B.

7) What is the Frobenius norm used for?

It summarizes the matrix magnitude after numeric evaluation by squaring entries, summing them, and taking the square root.

8) Will CSV and PDF exports keep the result section?

Yes. CSV exports the computed table or scalar. PDF captures the visible result card, including the summary and graph.