3x3Matrix

A 3x3 matrix is a square array with three rows and three columns. It is commonly denoted A or M, with entries aij where i designates the row and j the column (i, j ∈ {1, 2, 3}). Three-by-three matrices are used to represent linear transformations from R3 to R3 and to encode systems of three linear equations in three variables.

Operations on 3x3 matrices include addition and subtraction performed element-wise, scalar multiplication that scales every entry, and matrix multiplication defined by (AB)ij = sum over k of aik bkj. The transpose of a 3x3 matrix is formed by swapping rows and columns. The determinant, det(A), is a scalar that measures volume scaling under the associated linear transformation and determines invertibility: A is invertible if det(A) ≠ 0.

For a 3x3 matrix, the determinant can be computed by the rule of Sarrus or by expansion of minors: det(A) = a11(a22a33 − a23a32) − a12(a21a33 − a23a31) + a13(a21a32 − a22a31). If det(A) ≠ 0, the inverse A−1 exists and satisfies AA−1 = A−1A = I. The eigenvalues of A are roots of the characteristic polynomial det(A − λI) = 0, which is a cubic equation and may yield real or complex numbers.

Special forms include diagonal, upper triangular, and lower triangular matrices. Orthogonal matrices satisfy AT A = I, with det ±1; rotation matrices in three dimensions are orthogonal with det = 1. 3x3 matrices are widely used in solving linear systems, 3D graphics transformations, physics, and engineering analyses.