Matrix Inverse Calculator

Matrix Inverse is evaluated from Matrix A[1,1], Matrix A[1,2] and Matrix A[2,1]. The calculation reports Determinant, A⁻¹[1,1] and A⁻¹[1,2].

Results

Thanks — we’ve logged this for review.

About the Matrix Inverse Calculator

Matrix Inverse is treated here as a quantitative relation between Matrix A[1,1], Matrix A[1,2], Matrix A[2,1] and Matrix A[2,2] and Determinant, A⁻¹[1,1], A⁻¹[1,2] and A⁻¹[2,1].

The calculator uses a multi formula configuration. Each reported value is read as a direct evaluation of the stored rules with the declared field formats and units.

Formula basis:
Swap main diagonal elements (a and d), negate off-diagonal elements (b and c), divide all by the determinant. Result is the unique matrix satisfying A x A⁻¹ = I. If determinant = 0, the matrix is singular (non-invertible).

Interpret the outputs in the order shown by the result fields. Optional inputs affect only the outputs that depend on those variables.

Formula & How It Works

The calculation applies the following relations exactly as recorded in the metadata:

Swap main diagonal elements (a and d), negate off-diagonal elements (b and c), divide all by the determinant. Result is the unique matrix satisfying A x A⁻¹ = I. If determinant = 0, the matrix is singular (non-invertible).

Each output field is produced by substituting the supplied inputs into the relevant relation and then applying the declared rounding or text format.

Worked Examples

Example 1: Standard 2×2 inverse: [[4,7],[2,6]]

Inputs

a11: 4 a12: 7 a21: 2 a22: 6
Determinant: 10. A⁻¹[1,1]: 0.6. A⁻¹[1,2]: -0.7. A⁻¹[2,1]: -0.2. A⁻¹[2,2]: 0.4. Invertible?: Yes - matrix is invertible

With Matrix A[1,1] = 4, Matrix A[1,2] = 7, Matrix A[2,1] = 2 and Matrix A[2,2] = 6 as the stated inputs, the result is Determinant = 10, A⁻¹[1,1] = 0.6 and A⁻¹[1,2] = -0.7. Each value corresponds to the declared output fields.

Example 2: Identity-adjacent matrix: [[2,1],[1,1]]

Inputs

a11: 2 a12: 1 a21: 1 a22: 1
Determinant: 1. A⁻¹[1,1]: 1. A⁻¹[1,2]: -1. A⁻¹[2,1]: -1. A⁻¹[2,2]: 2. Invertible?: Yes - matrix is invertible

With Matrix A[1,1] = 2, Matrix A[1,2] = 1, Matrix A[2,1] = 1 and Matrix A[2,2] = 1 as the stated inputs, the result is Determinant = 1, A⁻¹[1,1] = 1 and A⁻¹[1,2] = -1. Each value corresponds to the declared output fields.

Example 3: Near-singular matrix: [[1,2],[2,4]]

Inputs

a11: 1 a12: 2 a21: 2 a22: 4
Determinant: 0. A⁻¹[1,1]: 0. A⁻¹[1,2]: 0. A⁻¹[2,1]: 0. A⁻¹[2,2]: 0. Invertible?: No - singular matrix (det = 0)

With Matrix A[1,1] = 1, Matrix A[1,2] = 2, Matrix A[2,1] = 2 and Matrix A[2,2] = 4 as the stated inputs, the result is Determinant = 0, A⁻¹[1,1] = 0 and A⁻¹[1,2] = 0. Each value corresponds to the declared output fields.

Example 4: Rotation matrix: 90 degrees, [[0,-1],[1,0]]

Inputs

a11: 0 a12: -1 a21: 1 a22: 0
Determinant: 1. A⁻¹[1,1]: 0. A⁻¹[1,2]: 1. A⁻¹[2,1]: -1. A⁻¹[2,2]: 0. Invertible?: Yes - matrix is invertible

With Matrix A[1,1] = 0, Matrix A[1,2] = -1, Matrix A[2,1] = 1 and Matrix A[2,2] = 0 as the stated inputs, the result is Determinant = 1, A⁻¹[1,1] = 0 and A⁻¹[1,2] = 1. Each value corresponds to the declared output fields.

Common Use Cases

  • Find inverse of 2×2 matrix for solving linear systems
  • Check if a matrix is invertible (non-zero determinant)
  • Solve Ax = b by computing x = A⁻¹b