3x3matris

3x3matris, or a 3x3 matrix, is a square array with three rows and three columns. It is typically denoted A = [a11 a12 a13; a21 a22 a23; a31 a32 a33], where aij is the element in row i and column j. In linear algebra, a 3x3 matrix represents a linear transformation from R^3 to R^3 relative to the standard basis, acting on a column vector x as Ax.

The determinant det(A) is a single number that determines invertibility: A is invertible if and only if det(A) ≠ 0. For 3x3 matrices, det can be computed by the rule of Sarrus: det(A) = a11 a22 a33 + a12 a23 a31 + a13 a21 a32 − a13 a22 a31 − a12 a21 a33 − a11 a23 a32. The determinant also relates to volume scaling under the transformation represented by A.

The trace tr(A) = a11 + a22 + a33 is the sum of diagonal entries and relates to the matrix's eigenvalues. Eigenvalues λ satisfy det(A − λI) = 0, and A can be diagonalized when there exists a full set of eigenvectors. Special classes include rotation matrices, which are orthogonal with det = 1, and symmetric matrices, which have real eigenvalues and orthogonal eigenvectors.

Inversion and computation: If det(A) ≠ 0, the inverse A^{-1} exists and can be computed via the adjugate or Gaussian elimination. Numerical methods often compute LU decomposition A = LU to solve systems and invert efficiently.

Applications: 3x3 matrices appear in solving systems of three linear equations, in 3D computer graphics for transforming coordinates and normals, and in physics for representing linear transformations and rotations in space.

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