Matrix Calculator (2×2)

Calculate determinant, inverse, transpose, and multiplication for 2×2 matrices.

By Konstantin Iakovlev · Updated September 2026 · Source: Khan Academy

Matrix A (2×2)

Operation

Determinant

-2.0000

Details

Determinant of A-2.0000

Use the Matrix Calculator (2×2) above to calculate your results. Enter your values and see instant results — all calculations run in your browser.

Disclaimer: This calculator is for informational purposes only. Results are estimates based on the information you provide and the assumptions described on this page.

How It Works

Linear transformations and systems of equations live and die on matrix arithmetic, which is why students, engineers, and data scientists keep this kind of tool close at hand. The same operations sit underneath computer graphics, statistics, and machine learning.

Enter a 2x2 matrix in the form [[a, b], [c, d]] and the calculator returns its determinant (ad-bc), its inverse (1/det * [[d, -b], [-c, a]]), its transpose ([[a, c], [b, d]]), and its product with a second 2x2 matrix using the standard row-by-column multiplication rule. Each result follows directly from the entries you supply.

Before computing an inverse, check that the determinant is not zero. A determinant of zero marks a singular matrix, which has no inverse at all. The other pitfall to watch for is sign placement: drop or misplace a negative during the inverse step and the transformation it represents will come out wrong, whether you are working in geometry or data analysis.

Example: Inverting a Feature-Scaling Matrix

  1. 1 A data pipeline transforms two features with the matrix M = [[2.5, 0.8], [0.3, 1.2]], and you need its inverse to map transformed values back. Enter a₁₁ = 2.5, a₁₂ = 0.8, a₂₁ = 0.3, a₂₂ = 1.2 and choose Inverse.
  2. 2 Determinant: (2.5 × 1.2) − (0.8 × 0.3) = 3.0 − 0.24 = 2.76. It is not zero, so the inverse exists: (1/2.76) × [[1.2, −0.8], [−0.3, 2.5]].
  3. 3 The calculator shows the inverse [[0.4348, -0.2899], [-0.1087, 0.9058]] and the determinant of A, 2.7600.
  4. 4 As a check, M × M⁻¹ is the identity matrix. For the top-left entry: 2.5 × (1.2/2.76) + 0.8 × (−0.3/2.76) = (3.0 − 0.24) / 2.76 = 1.

Source: Khan Academy · Last updated: September 2026

Frequently Asked Questions

How do you find the determinant of a 2x2 matrix?
For a matrix [[a,b],[c,d]], the determinant is ad - bc. For example, [[3,2],[1,4]] has determinant (3x4) - (2x1) = 12 - 2 = 10.
How do you find the inverse of a 2x2 matrix?
For matrix [[a,b],[c,d]], the inverse is (1/determinant) x [[d,-b],[-c,a]]. The matrix must have a nonzero determinant (otherwise it has no inverse).
How do you multiply two 2x2 matrices?
Multiply rows of the first matrix by columns of the second. For [[a,b],[c,d]] x [[e,f],[g,h]]: top-left = ae+bg, top-right = af+bh, bottom-left = ce+dg, bottom-right = cf+dh.

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