2x2 Matrix Calculator

A 2x2 matrix is an arrangement of numbers in 2 rows and 2 columns. It is the simplest example of a square matrix used in linear algebra.

Matrix multiplication proceeds row by column. Element c11 = a11*b11 + a12*b21, c12 = a11*b12 + a12*b22, c21 = a21*b11 + a22*b21, c22 = a21*b12 + a22*b22.

The determinant of a 2x2 matrix is a scalar value computed as det(A) = a11*a22 - a12*a21. It determines, among other things, whether the matrix is invertible.

A matrix is singular (non-invertible) when its determinant equals 0. This means the system of linear equations described by the matrix has no unique solution.

No, matrix multiplication is not commutative. In general A*B ≠ B*A. The order of factors in matrix multiplication is essential to the result.

Matrices are added element by element: each element of the result matrix is the sum of the corresponding elements of the component matrices. Both factors must have the same dimensions.

The 2x2 identity matrix has 1s on the diagonal and 0s elsewhere: [[1,0],[0,1]]. It is the neutral element of matrix multiplication — A*I = I*A = A.

For matrix [[a,b],[c,d]] the inverse is (1/det)*[[d,-b],[-c,a]], where det = a*d - b*c. The inverse exists only when the determinant is non-zero.

Matrices are used in computer graphics (3D transformations), cryptography, statistics, quantum physics, machine learning and for solving systems of linear equations.

Transposition swaps rows and columns (a12 swaps with a21), while the inverse matrix satisfies A*A⁻¹ = I. These are different mathematical operations.