2x2 Inverse Matrix Calculator

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Calculate the inverse of a 2×2 square matrix. An inverse matrix A^{-1} is such that A · A^{-1} = I.

Matrix Configuration

Matrix Inverse Result

3Determinant

Determinant

Inverse Matrix

0.667

-0.333

0.667

Solution Steps

#1Calculate determinant: D = ad - bc = (2)(2) - (1)(1) = 3.

#2Swap main diagonal elements a=2 and d=2.

#3Change signs of b=1 and c=1.

#4Final matrix: A^{-1} = \frac{1}{3} \begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix}.

Invertible?Yes

TypeSquare 2x2

About this calculator

Matrix Inversion (2x2)

The inverse of a 2×2 matrix exists only if its determinant is non-zero. For a matrix A = [[a,b],[c,d]], the inverse is calculated by swapping the main diagonal elements (a and d), negating the off-diagonal elements (b and c), and multiplying the resulting matrix by 1/det(A).

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Pro Tips

Check Determinant: If det(A) = 0, the matrix is singular and has no inverse.
Identity Check: Multiplying a matrix by its inverse results in the identity matrix I.
Signs: Don't forget to flip signs of b and c when forming the adjugate matrix.

Fun Facts

"A matrix with no inverse is called 'singular' or 'degenerate'."
"Matrix inversion is a core part of linear algebra used in computer graphics and machine learning."
"Inverse matrices are used to solve systems of linear equations in matrix form (Ax = b)."

Formula: A^{-1} = (1/(ad-bc)) [[d,-b],[-c,a]]