Matrix Inverse Calculator
Calculate the inverse of a square matrix used in linear algebra operations.
About Matrix Inverse Calculator
The inverse of a 2×2 matrix [[a,b],[c,d]] is (1/det) × [[d,−b],[−c,a]] where det = ad − bc. A matrix is invertible only if its determinant is non-zero. Matrix inversion is used in solving linear systems, computing covariance matrices, and certain optimization algorithms in ML.
Formula
2×2 Inverse
A^(-1) = (1/(ad-bc)) * [[d, -b], [-c, a]]
- a,b,c,d
- Elements of the 2×2 matrix
Worked example
With the default values below, Matrix Inverse Calculator returns Determinant of -2.000000. Change any input above and every figure here is recalculated in your browser.
Inputs used
- a (row1,col1)
- 1
- b (row1,col2)
- 2
- c (row2,col1)
- 3
- d (row2,col2)
- 4
Results
- Determinant
- -2.000000
- Inv[1,1]
- -2.000000
- Inv[1,2]
- 1.000000
- Inv[2,1]
- 1.500000
- Inv[2,2]
- -0.500000
Step by step
- 1Determinant: ad − bc= -2.000000
- 2Invertible?= Yes
Assumptions and limits
- The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 6 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
Why avoid explicit matrix inversion in practice?
Direct inversion is numerically unstable and O(n³). In ML, solving Ax=b via LU or Cholesky decomposition is preferred, and most optimization algorithms use iterative methods instead of computing inverses.
What does a singular matrix mean in ML?
A singular (non-invertible) matrix has linearly dependent rows/columns, indicating redundant features. This can cause issues in algorithms like linear regression if the feature matrix is rank-deficient.