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Inverse of a Matrix

Matrix inverse calculator. Enter A to compute A⁻¹ by row reduction and verify AA⁻¹=I—for linear algebra.

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A·A⁻¹ = A⁻¹·A = I
space-separated entries, semicolon-separated rows, e.g. 1 2;3 4

📖 Tutorial｜Inverse of a Matrix

1. Definition
If B satisfies AB = BA = I, B is the inverse of A, denoted A⁻¹.
2. Symbols
SymbolMeaning
A⁻¹inverse of A
Iidentity matrix
det Adeterminant; 0 means singular
3. How this tool works
  • A square matrix is invertible iff det A ≠ 0;
  • Augment [A|I] and row-reduce; when left becomes I, right is A⁻¹;
  • This tool computes the inverse and verifies A·A⁻¹=I.
4. Steps
  1. Enter A;
  2. Click Calculate to see A⁻¹;
  3. If det A=0 it reports no inverse.
5. Example
Example: A=4 7;2 6. det=24−14=10≠0, A⁻¹=0.6 −0.7;−0.2 0.4, and A·A⁻¹=I.
6. Pitfalls
Non-square matrices have no inverse;
A singular matrix (det 0) has no inverse;
The inverse is not the elementwise reciprocal.

❓ FAQ｜Inverse of a Matrix

Which matrices have an inverse?
Square matrices with det A ≠ 0 (invertible/non-singular).
How to compute the inverse?
Gauss-Jordan: row-reduce [A|I] to [I|A⁻¹].
What if det A=0?
The matrix is singular and has no inverse.
Is there a 2×2 formula?
[[a,b],[c,d]]⁻¹ = (1/det)·[[d,−b],[−c,a]].
Is the inverse unique?
Yes.