How to find 2x2 inverse?

Swap a,d. Negate b,c. Divide by det(ad-bc). The shortcut formula for 2x2.

When no inverse exists?

When det=0 (singular). Rows/columns are linearly dependent.

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Inverse Matrix Calculator

Compute A^(-1) for 2x2 and 3x3 matrices step by step

Enter 2x2 Matrix

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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