Matrix Inverse Calculator
Calculate the inverse of a 2×2 or 3×3 matrix and its determinant.
Inputs
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
Determinant: ad − bc
1×4 − 2×3
= -2.000000
Invertible?
|det| > 0
= Yes
How it works
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
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.