Summary On Special matrices and Elementary row Operations of Matrices
- any square matrix A is said to be symmetric if it is equal to its transposei.e. $A = A^{T}$
- Any square matrix A is said to be skew-symmetric if $A = – A^{T}$ or $A=A^{T}=0$.
- An elementary operation of matrices is a simple operation made on rows or columns which aims transforming any matrix to row or column equivalent matrix.
- Swapping:- interchanging rows of a matrix i.e. $R_i \leftrightarrow R_j$
- Re-scaling:- multiplying a row of a matrix by anon zero constant: $R_i \rightarrow rR_i$
- Pivoting :- adding constant multiple of one row of a matrix on to another row. $R_i \rightarrow R_i + rR_j$
- A matrix is said to be in row echelon form if: A zero row( if there is ) comes at the bottom. The first non-zero entry in each non-zero row is 1. The number of zero entries down the row is increasing.
- A matrix is said to be row reduced echelon (RREF) if and only if: It is in echelon form. The first non-zero entry in each non-zero row is the only non-zero entry in its column.