2x2 Matrix Inverse Calculator

ADJUGATE / IDENTITY CHECK

Turn four matrix entries into an inverse you can verify by multiplication

This explainer is designed for the part of a 2x2 inverse that usually gets split across several result rows. With the reviewed default matrix [[4, 7], [2, 6]], the calculator first finds det(A) = 10, then rescales the adjugate [[6, -7], [-2, 4]] by 1/10 to return A^-1 = [[0.6, -0.7], [-0.2, 0.4]]. The visual below keeps the swap-and-negate step and the identity check in one place.

Worked default example

Entered matrix

A = [[4, 7], [2, 6]], where a = 4, b = 7, c = 2, and d = 6.

Invertibility check

The calculator first checks det(A) = ad - bc. Here 4 x 6 - 7 x 2 = 10, so the matrix is invertible.

Adjugate step

Swap the diagonal entries and negate the off-diagonal entries to get adj(A) = [[6, -7], [-2, 4]].

Scale every entry by 1 / det(A)

1/10 x 6 = 0.6; 1/10 x (-7) = -0.7; 1/10 x (-2) = -0.2; 1/10 x 4 = 0.4.

Returned inverse matrix

Read the main result and the three supporting metrics together: A^-1 = [[0.6, -0.7], [-0.2, 0.4]].

What to enter and how to read the result

Enter the four scalar entries in row order: top-left a, top-right b, bottom-left c, and bottom-right d. The main result card reports inverse entry (1,1), and the three metrics below it report entries (1,2), (2,1), and (2,2). Read all four together as the inverse matrix for the same 2x2 input.

What the calculator actually does

Nirmion evaluates A^-1 = (1 / (ad - bc))[[d, -b], [-c, a]] directly from the entered entries. In the reviewed default example, det(A) = 10, the adjugate is [[6, -7], [-2, 4]], and multiplying A by [[0.6, -0.7], [-0.2, 0.4]] reproduces the identity matrix [[1, 0], [0, 1]] through the four row-by-column checks shown in the audit table.

Where this model stops helping

This page handles real-number 2x2 entries only and requires a nonzero determinant. It does not keep symbolic fractions, invert larger matrices, diagnose near-singularity, or multiply the inverse by another matrix or constants vector for you. If det(A) = 0, the matrix is singular and has no inverse.

Reference and next checks

The linked OpenStax section states the 2x2 inverse formula and shows how an inverse is validated against the identity matrix. If you want to inspect the determinant first, solve a full two-equation system, or compare eigenvalues from the same matrix next, use the related calculators below.