Matrices | Class 12 Mathematics Notes

3.6 Symmetric and Skew Symmetric Matrices

A square matrix A is symmetric if it equals its transpose, i.e., A' = A, meaning a_ij = a_ji for all i and j. A square matrix A is skew symmetric if its transpose equals the negative of itself, i.e., A' = -A, implying a_ji = -a_ij. Consequently, all diagonal elements of a skew symmetric matrix are zero since a_ii = -a_ii implies a_ii = 0. Theorems state that for any square matrix A, the matrix A + A' is symmetric, and A - A' is skew symmetric. Moreover, any square matrix can be expressed uniquely as the sum of a symmetric and a skew symmetric matrix: A = 1/2 (A + A') + 1/2 (A - A'). This decomposition is fundamental in matrix theory and applications.

🧪 Activity: Example 22: Decomposing matrix B into symmetric and skew symmetric parts.

🔗 Connection: Leads to study of invertible matrices and matrix inverses.