MCPCalc

Matrix Calculator

Operations on 2×2 to 4×4 matrices: addition, multiplication, determinant (with cofactor expansion shown), inverse, RREF, rank, and more.

Result:

[3, 7]
[2, 8]

Need eigenvalues, LU decomposition, nullspace, or symbolic manipulation? Continue in CAS workspace →

What This Calculator Does

Compute determinant, inverse, and eigenvalues for a 2x2 matrix.

It combines a11, a12, a21, a22 to estimate Determinant, Inverse, Eigenvalue 1.

Formula & Method

Core equations: For A=[abcd]A=\begin{bmatrix}a&b\\c&d\end{bmatrix}: det⁡A=ad−bc\det A=ad-bc, A−1=1ad−bc[d−b−ca]A^{-1}=\frac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}, eigenvalues from λ2−tr(A)λ+det⁡(A)=0\lambda^2-\text{tr}(A)\lambda+\det(A)=0.

Notation used in the formulas: RR = Determinant; x1x_{1} = a11; x2x_{2} = a12; x3x_{3} = a21; x4x_{4} = a22.

Method summary: inputs are normalized to consistent units, core equations are evaluated, then secondary values are derived and rounded for display.

Use this calculator for quick scenario analysis. Start with baseline values, change one driver at a time, and compare how sensitive the results are to each input shown above.

Reference Book

Linear Algebra and Its Applications

David C. Lay, Judi J. McDonald, and Steven R. Lay · Pearson

A standard text for linear systems, matrices, eigenvalues, diagonalization, and matrix factorizations.

View Book

Inputs Used

  • a11: Used directly in the calculation.
  • a12: Used directly in the calculation.
  • a21: Used directly in the calculation.
  • a22: Used directly in the calculation.

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Related Learning

Use these guided proofs, examples, and explanations to go deeper on the ideas behind this tool.

  • What a determinant measures geometrically

    The determinant of a 2 by 2 matrix is the signed area scale factor of the transformation. It tells whether the unit square stretches, flips, or collapses.

  • How a linear transformation changes the plane

    A linear transformation of the plane is completely determined by where it sends the two standard basis vectors. From there, every grid point follows by the same linear combination.

  • What a basis does in linear algebra

    A basis is what turns a vector space into something you can navigate. It reaches every vector, and it does so without redundancy, so coordinates become possible.

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