Inverse Matrix Calculator
Compute A^(-1) for 2x2 and 3x3 matrices step by step
2x2
3x3
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Inverse Matrix Formula
A^(-1) = (1/det(A)) x adj(A)
2x2: [[a,b],[c,d]]^-1 = 1/(ad-bc) x [[d,-b],[-c,a]]
Condition: det(A) != 0
A x A^(-1) = I
The inverse matrix solves Ax=b. Only square non-singular matrices are invertible. The adjugate method works for any size.
⚠ A matrix is invertible only if det != 0.
What is Matrix Inverse?
The inverse A^-1 is the unique matrix where A x A^-1 = I. It is the matrix analogue of the reciprocal.
2x2 Shortcut
Swap a,d. Negate b,c. Divide by det.
Adjugate
Minors, cofactors, transpose, divide by det.
Properties
(AB)^-1 = B^-1 A^-1, det(A^-1) = 1/det(A).
Verify
A x A^-1 should equal identity matrix I.
Example: [[4,7],[2,6]]. det=10. adj=[[6,-7],[-2,4]]. A^-1 = [[0.6,-0.7],[-0.2,0.4]].
Applications
Linear Systems
Graphics
Least Squares
Control
Frequently Asked Questions
What is an inverse matrix?▼
A x A^-1 = I. Only square non-singular matrices have inverses.
2x2 inverse?▼
Swap a,d. Negate b,c. Divide by det(ad-bc).
No inverse?▼
When det=0 (singular matrix), no inverse exists.
How to find 3x3 inverse?▼
Minors, cofactors, transpose, divide by det.
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