Matrix Algebra

Row Equivalence

Row Equivalence. The two systems of linear equations

$\displaystyle \left\{\begin{array}{rrrr} x_1 & -2 x_2 & + x_3 = & 0 \\ & 2 x_2 & -8 x_3 = & 8 \\ -4 x_1 & +5 x_2 & +9 x_3 = & -9 \end{array}\right.$ and $\displaystyle \quad \left\{\begin{array}{rrrr} x_1 & -2 x_2 & + x_3 = & 0 \\ & 2 x_2 & -8 x_3 = & 8 \\ & -3 x_2 & +13 x_3 = & -9 \end{array}\right.$

are equivalent in the sense that they should have the same solution. Equivalently by applying the row operations $4 \times R_1 + R_3 \to R_3$ in the augmented matrix, we obtain the corresponding matrices

$\displaystyle \left[\begin{array}{rrrr} 1 & -2 & 1 & 0 \\ 0 & 2 & -8 & 8 ... ... 1 & -2 & 1 & 0 \\ 0 & 2 & -8 & 8 \\ 0 & -3 & 13 & -9 \end{array}\right]$

The symbol “ $\sim$ ” indicates that the two matrices are row equivalent.