3×3matrix

A 3×3 matrix is an array with three rows and three columns whose entries come from a field, typically the real or complex numbers. It is usually written as A = [a11 a12 a13; a21 a22 a23; a31 a32 a33], where aij denotes the entry in row i and column j. 3×3 matrices are used to represent linear transformations from three-dimensional space to itself and can be used to model systems of three equations in three unknowns.

Core operations include addition, subtraction, and scalar multiplication performed elementwise, as well as matrix multiplication, which combines two linear transformations. The transpose AT of a matrix A is obtained by swapping rows and columns. A 3×3 matrix is invertible exactly when its determinant det(A) is nonzero; in that case it has a unique inverse A−1 satisfying AA−1 = A−1A = I3.

The determinant of a 3×3 matrix can be computed by a standard formula: det(A) = a11(a22a33 − a23a32) − a12(a21a33 − a23a31) + a13(a21a32 − a22a31). The eigenvalues λ are solutions of det(A − λI) = 0, and corresponding eigenvectors describe directions that are scaled by A. Invertible 3×3 matrices preserve linear independence and volume up to a factor equal to the determinant.

Special types include diagonal matrices, where off-diagonal entries are zero; orthogonal matrices, for which AT = A−1 and whose determinant is ±1; and rotation or reflection matrices used in 3D graphics. A 3×3 matrix can also be used to describe affine transformations when combined with homogeneous coordinates.

Applications span solving linear systems, computer graphics, physics, and engineering. A concrete 3×3 example has determinant 22, indicating the matrix is invertible and has a well-defined inverse.