Matrix Algebra

Row Operations

Rows and Columns. Let $A = \begin{bmatrix} 1 & -2 & 1 0 & 2 & -8 -4 & 5 & 9 \end{bmatrix}$ be a matrix. Then $[\: 1\: {-2}\:\: 1\:]$ , $[\:0 \:\:2 \:{-8} \:]$ , and $[\:{-4} \:\:5 \:\:9 \:]$ are respectively called the first, the second, and the third row, and denoted by , , and . Similarly $\begin{bmatrix}1 0 -4 \end{bmatrix}$ , $\begin{bmatrix}-2 2 5 \end{bmatrix}$ , and $\begin{bmatrix}1 -8 9 \end{bmatrix}$ are respectively called the first, the second, and the third column of the matrix .

Row Operatoins. The following three basic row operations are used to systematically produce a reduced echelon form:

Replacement: $k \times R_i + R_j \to R_j$ (add a multiple of the -th row by to the -th row)
Interchange: $R_i \leftrightarrow R_j$ (interchange the -th row and the -th row)
Scaling: $k \times R_i \to R_i$ (multiply the -th row by )

Given a matrix A in Matlab/Octave, the above three row operations are carried out as follows:

A(j,:) = A(j,:) + k * A(i,:)
A([j,i],:) = A([i,j],:)
A(i,:) = k * A(i,:)