2x2 Adjugate Calculator

Actual2×2classical adjugate[[d,−b],[−c, a]], even for singular matrix, not conjugate transpose or inverse.

Description

The classical adjugate is the transpose of the cofactor matrix and remains defined for singular matrices.

The classical adjugate is the transpose of the cofactor matrix and remains defined for singular matrices.^[1]

This tool returns the 2-by-2 adjugate without dividing by a determinant or taking a conjugate transpose.

When to use 2x2 Adjugate Calculator

Use this tool to obtain the 2-by-2 adjugate without dividing by a determinant or taking a conjugate transpose.

How to use 2x2 Adjugate Calculator and how it works

Prepare actualMatrix, with the units, conventions and assumptions below.
Enter the required fields and run 2x2 Adjugate Calculator.
Read result.adjugate using the output definitions below.
Correct any rejected inputs and rerun; interpret the result within the stated model and its assumptions.

Input field	Meaning	Value type	Requirement
actualMatrix	Exactly two rows of two finite entries, using the row and column ordering below.	array of array of number	Required; other constraints below

Require a finite 2-by-2 matrix [[a,b],[c,d]]. Return [[d,-b],[-c,a]], the transpose of its cofactor matrix. Singular matrices are accepted because the adjugate exists even when an inverse does not.

adjugate([[a,b],[c,d]])=[[d,-b],[-c,a]]

Apply the input ordering, units and branch conditions stated above.

Output field	Meaning	Value type
result	Object containing the calculated or selected fields below.	object
result.adjugate	2-by-2 adjugate preserving the stated row and column order.	array of array of number

json

{
  "input": {
    "actualMatrix": [
      [
        1,
        2
      ],
      [
        3,
        4
      ]
    ]
  },
  "output": {
    "result": {
      "adjugate": [
        [
          4,
          -2
        ],
        [
          -3,
          1
        ]
      ]
    }
  }
}

Limitations and assumptions

The calculation uses the supplied observations and model inputs. It does not collect measurements, select a reference population or coefficient, or establish that the model applies to the actual case. The stated branch conditions and output units remain part of the result.

All fields require explicit values. Unexpected fields, wrong types, nonfinite numbers and sparse arrays are rejected. Alignment, bounds and supported-token rules are explained above; extremely large inputs or nonfinite results can also fail. Input errors differ from successful false comparisons, unavailable-domain flags and empty match lists.

Alternative or Complementary methods

Use 2x2 Eigenvectors Calculator for a complementary operation: A real matrix can have real, repeated or complex eigenvalues, and repeated values do not ensure diagonalizability.

Use 2x2 Condition Number Calculator for a complementary operation: A spectral condition number compares the largest and smallest singular values of a matrix.

References

2x2 Adjugate Calculator — tool.io

Similar or alternative tools

2x2 Eigenvectors Calculator
A real matrix can have real, repeated or complex eigenvalues, and repeated values do not ensure diagonalizability.
2x2 Condition Number Calculator
A spectral condition number compares the largest and smallest singular values of a matrix.