Inverse Matrix Calculator
Calculate the inverse of a 2x2 or 3x3 matrix
Matrix Inverse Calculator – Find A⁻¹ of Any Matrix
Calculate the inverse of any square matrix up to 10×10 with our free online matrix inverse calculator. Enter values via textarea or row-by-row input with step-by-step solutions.
Current matrix: 2×2 | Detected from your input
What is a Matrix Inverse?
The inverse of a square matrix A, denoted A⁻¹, is a matrix such that when multiplied by A gives the identity matrix: A × A⁻¹ = A⁻¹ × A = I. Not all matrices have inverses – only non-singular (invertible) matrices with non-zero determinants.
A matrix with determinant 0 is called singular and has no inverse.
Matrix Inverse Formulas
2×2 Matrix Inverse
A = [a b]
[c d]
A⁻¹ = (1/det(A)) × [d -b]
[-c a]Swap diagonal elements, negate off-diagonal, divide by determinant
General Formula
A⁻¹ = (1/det(A)) × adj(A)
where adj(A) is the adjugate
(transpose of cofactor matrix)Works for any square matrix
Properties of Matrix Inverses
Key Properties
- • (A⁻¹)⁻¹ = A
- • (AB)⁻¹ = B⁻¹A⁻¹
- • (Aᵀ)⁻¹ = (A⁻¹)ᵀ
- • (kA)⁻¹ = (1/k)A⁻¹
- • A × A⁻¹ = A⁻¹ × A = I
When Inverse Exists
- • Matrix must be square
- • Determinant must be non-zero
- • Rows/columns must be independent
- • Matrix must be full rank
Frequently Asked Questions
What does it mean if a matrix has no inverse?
A matrix with no inverse is called singular. This happens when the determinant is 0, meaning the rows or columns are linearly dependent. Such matrices cannot be used to solve systems of equations uniquely.
How do you verify an inverse is correct?
Multiply the original matrix A by the calculated inverse A⁻¹. If the result is the identity matrix (1s on diagonal, 0s elsewhere), the inverse is correct: A × A⁻¹ = I.
Can non-square matrices have inverses?
No, only square matrices can have true inverses. However, rectangular matrices can have pseudo-inverses (Moore-Penrose inverse) used in least squares problems.
What is the adjugate matrix?
The adjugate (or adjoint) matrix is the transpose of the cofactor matrix. It's used in the formula A⁻¹ = (1/det(A)) × adj(A) to compute the inverse.
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