Example Find the rank of the matrix $A$.．

$$A = \left(\begin{array}{rrrr} 1&-2&5&3\\ 2&3&1&-1\\ 3&8&-3&-5 \end{array}\right)$$

Answer Note that the rank of matrix is the same as the nonzero rows of the reduced matrix.

$$A = \left(\begin{array}{rrrr} 1&-2&5&3\\ 2&3&1&-1\\ 3&8&-3&-5 \end{... ...left(\begin{array}{rrrr} 1&-2&5&3\\ 0&7&-9&-7\\ 0&14&-18&4 \end{array}\right)$$

$$\stackrel{\begin{array}{cc} {}^{\frac{1}{7} \times R_{2}}\\ {}^{-2R_{2} + R_{3}} \end{array}}{\longrightarrow} \left(\begin{array}{rrrr} 1&-2&5&3\\ 0&1&\frac{-9}{7}&-1\\ 0&0&... ...rrrr} 1&0&\frac{17}{7}&1\\ 0&1&\frac{-9}{7}&-1\\ 0&0&0&0 \end{array}\right) .$$

Then the number of nonzero rows of the reduced matrix $A_{R}$ is $2$. Thus ${\rm rank}(A) = 2$. Furthermore, the row vector $${(1,0,\frac{17}{7},1)}$$ of $A_{R}$ and the row vector $${(0,1,\frac{-9}{7},-1)}$$ are bases of vector space $A$. Therefore, the dimension of the row space is $2$．