Inverse Matrix Calculator (2×2)

A square matrix has an inverse exactly when its determinant is not zero, and this calculator finds it for 2×2, 3×3, 4×4, and 5×5 matrices. The 2×2 path shows the classic formula step by step (determinant, adjugate, divide), larger sizes use Gauss-Jordan elimination, and every result is verified by multiplying A x A^-1 back to the identity matrix.

Quick Answer

How to use the inverse matrix calculator

Pick the size, type the entries row by row, and click Calculate. Results stay hidden until you click.

1. Matrix size: 2×2 is the default and is pre-filled with the worked example [[4, 7], [2, 6]]. Choosing 3×3, 4×4, or 5×5 rebuilds the grid; the larger grids start blank, with the entry positions labelled a11, a12, and so on.
2. Matrix A entries: enter every entry, reading across rows. Decimals and negatives are fine; a blank or non-numeric entry stops the calculation with a message naming the exact entry to fix.
3. Read the result: the inverse is the headline. Below it sit the determinant, the verification product, the full step-by-step working with your numbers, and a one-line verdict you can copy with the Copy button.
4. Singular matrices: if the determinant is zero, the inverse line reads “does not exist” and the verdict explains why, rather than returning an error or a garbage matrix.

The 2×2 inverse formula: determinant, adjugate, divide, verify

For A = [[a, b], [c, d]], the inverse is A^-1 = (1 / (ad – bc)) x [[d, -b], [-c, a]], valid whenever ad – bc is not zero. The OpenStax Algebra and Trigonometry section on solving systems with inverses presents the same formula and the same condition: a matrix is invertible exactly when its determinant is non-zero.

Geometrically, the determinant is the signed area scale factor of the transformation, which is why zero is fatal: a matrix with determinant zero flattens the plane onto a line (or a point), and a flattening cannot be undone. A negative determinant, such as the -2 in the table below, is no problem at all: the inverse exists, and the negative sign records that the transformation flips orientation. The inverse is also the engine behind solving systems as x = A^-1b, and the same singularity question, whether (A – λI) can be inverted, sits underneath the eigen values calculator.

Worked example, plus four more verified runs

For A = [[4, 7], [2, 6]], the tool prints exactly this working, and every other row in the table is a verified run of the same calculator:

• Step 1, determinant: det(A) = 4×6 – 7×2 = 24 – 14 = 10
• Step 2, adjugate: swap the diagonal, negate the off-diagonal, giving [[6, -7], [-2, 4]]
• Step 3, divide by det: A^-1 = (1/10) x [[6, -7], [-2, 4]] = [[0.6, -0.7], [-0.2, 0.4]]
• Step 4, verify: A x A^-1 = [[1, 0], [0, 1]], the identity matrix

For exact fractions instead of decimals, read each inverse entry as an integer over the determinant before dividing: the worked example is [[6/10, -7/10], [-2/10, 4/10]]. The related eigen work on this site, such as the basis for eigenspace calculator and the eigenbasis calculator, leans on the same invertibility test, since an eigenspace is what remains when (A – λI) is singular.

3×3, 4×4, and 5×5: the Gauss-Jordan mode

Larger matrices invert by Gauss-Jordan elimination on the augmented matrix [A | I]: row operations reduce the left half to the identity, the right half becomes A^-1, and the determinant falls out as the product of the elimination pivots, with a sign flip for each row swap. The OpenStax section on solving systems with Gaussian elimination teaches the same row-reduction machinery this mode automates.

Verified 3×3 run: for A = [[2, 1, 1], [1, 2, 1], [1, 1, 2]], the determinant is 4 and the inverse is [[0.75, -0.25, -0.25], [-0.25, 0.75, -0.25], [-0.25, -0.25, 0.75]], with the verification product printing as the exact 3×3 identity. A diagonal 4×4 run, A = diag(2, 3, 4, 5), gives determinant 120 and the reciprocal diagonal [[0.5, 0, 0, 0], [0, 0.333333, 0, 0], [0, 0, 0.25, 0], [0, 0, 0, 0.2]]. Singular matrices are caught mid-elimination: [[1, 2, 3], [4, 5, 6], [7, 8, 9]] reports “does not exist” with determinant 0 because a pivot collapses to zero. Partial pivoting, always eliminating on the largest available pivot in the column, keeps rounding as small as this method allows.

Precision limits and the near-singular trap

Displayed numbers are rounded to 6 decimal places, and two honest consequences follow. First, repeating decimals truncate: the inverse of [[3, 0], [0, 3]] prints as 0.333333, not the exact 1/3. Second, a matrix can be invertible yet so close to singular that the display misleads: [[1, 1], [1, 1.0000001]] has a true determinant of about 0.0000001, which the display rounds to 0, and its step line even writes (1/0), yet because 0.0000001 sits above the tool’s singularity cutoff of 1e-12, an inverse is still returned, with entries near 10,000,000. Treat “determinant shows 0 but an inverse appears” as a warning, not a result: the matrix is nearly singular and tiny input changes swing that inverse wildly.

At the far extreme the display breaks down entirely: an entry such as 1e200 prints in the working as 1e+2, because the trailing-zero trim eats the exponent, and a determinant that overflows prints as Infinity alongside a zero inverse whose verification product is zeros, not the identity. Treat any output at that scale as out of range, not as an answer. The cutoffs themselves are part of the tool’s behaviour and are worth stating plainly. In 2×2 mode, any determinant with absolute value below 1e-12 is treated as zero and reported as singular, even if it is not exactly zero. In the larger sizes, the same 1e-12 threshold applies to elimination pivots, and a singular result prints the determinant as exactly 0. Verification uses a tolerance of 1e-9 in 2×2 mode and 1e-8 for larger sizes, so the product may show a “tiny floating-point rounding” note on extreme inputs instead of a clean pass. None of this affects ordinary homework or engineering matrices; it matters only at the numerical edge.

What competing calculators miss

Most inverse tools get the formula right; they differ on verification, size range, and whether the working teaches anything:

• eMathHelp’s inverse calculator shows full Gauss-Jordan steps, a real strength, but applies elimination even to 2×2 matrices, so the one-line adjugate pattern students are actually taught never appears, and there is no determinant-first readout or identity product in the same view.
• Symbolab’s matrix calculator covers inverses inside a general multi-operation tool with strong worked examples, but the inverse is one operation among many, steps sit behind its solver flow, and there is no built-in A x A^-1 verification line to catch a mistyped entry.
• Testbook’s inverse matrix calculator explains the cofactor method well in prose, including the common slip of writing 1/|A| where the signed 1/det(A) belongs, but the explanation and the tool are separate: the working is generic, not computed from your entries.
• Keisan-style inverse tools (such as the Luft/Keisan inverse matrix page) state the rules correctly (square matrix, non-zero determinant) and return fractions for 2×2 inputs, but stop at the answer matrix with no step trace and no verification product.
• The gap this page fills: determinant, adjugate steps in your numbers, the inverse, and the identity check on one screen for 2×2, Gauss-Jordan with pivot determinants for 3×3 to 5×5, a plain-language singular verdict, and the precision limits written down instead of hidden.

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