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Matrix Inverse Calculator

Find the inverse of a real two-by-two matrix and see the determinant used in the calculation. Enter the matrix in row order, then read the four inverse entries in the same order. A useful independent check is to multiply your original matrix by the displayed inverse; the result should be approximately the identity matrix, with ones on its main diagonal and zeros in the other positions. Keep extra precision for that check.

Enter every field, including explicit zeros. Decimal and scientific notation are accepted; commas, fractions and blank fields are rejected. Each input must be finite and no larger than 1 billion in absolute value.

Inverse row 1, column 1-2
Inverse row 1, column 21
Inverse row 2, column 11.5
Inverse row 2, column 2-0.5
Determinant-2

How to use this calculator

  1. Match each input to its labeled coordinate, coefficient or matrix position.
  2. Check the signs and use one consistent coordinate convention.
  3. Select Calculate, or press Enter, and read the labeled results.

Formula and method

For A = [[a,b],[c,d]], A⁻¹ = [[d,−b],[−c,a]]/(ad − bc).

The calculation runs locally in your browser. Changing an input requires selecting Calculate again. If an input or mathematical condition is invalid, the previous result is hidden instead of leaving a stale answer visible.

Worked example

For A with rows (1,2) and (3,4), the determinant is −2. Dividing the adjugate entries by −2 gives inverse rows (−2,1) and (1.5,−0.5).

Limitations and interpretation

Only real two-by-two matrices are supported. Matrices with zero determinant have no inverse. This tool also rejects |det| ≤ 10⁻¹² times the square of the largest absolute entry, because near-singular inputs can amplify rounding. That threshold is a numerical safeguard, not a formal classification theorem.

Results are displayed to approximately 10 significant digits. The underlying calculation uses ordinary floating-point numbers and is not exact symbolic arithmetic. Rounded results can differ slightly from hand calculations. For a measured quantity, choose precision that reflects the measurements rather than all the digits shown. Verify consequential calculations independently.

Frequently asked questions

Why is my nonzero determinant rejected?

A determinant very small relative to the entries can make inversion numerically unstable. Use higher-precision software for those cases.

Can I enter negative or decimal values?

Yes, provided the values meet the mathematical conditions described above. Type a normal hyphen for a negative sign. Do not type units, commas or a fraction such as 1/2; enter 0.5 instead.

Why did the calculator hide my previous result?

A missing or invalid input, undefined operation or unstable numerical case produces an explanation. Correct the input and calculate again to obtain a result for the current values.

Review notes and reference

Reviewed October 3, 2026. The stated default example and a boundary or algebraic check were tested against expected values. Input validation checks included blank fields, nonnumeric text and values outside the supported range. Wolfram MathWorld formula reference provides mathematical background. This tool is for calculation and education; its output alone does not validate a professional design.

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