I am reading about how to find the rank of matrices. And one way to find this is to see number of independent rows. As number of independent rows equals to the rank. If matrix is small i.e with 4 elments or 9. We can check linear independence. But if we have matrices with many elements then how can we check linear dependent rows. Is there any easy way?
Edit-
I know gaussian elimination method. But i want to know using linear depedence it is possible? If yes then easy method?
One method is the Gaussian elimination. For example:
$$\begin{aligned} A=&\begin{bmatrix}{\boxed{1}}&{2}&{-1}&2\\{4}&{1}&{0}&2\\{2}&{-3}&{2}&-2\\ 1&0&-1&3\\2&1&2&-4\end{bmatrix}\begin{matrix}\sim \\R_2-4R_1\\R_3-2R_1\\R_4-R_1\\R_5-RF_1\end{matrix}\\ &\begin{bmatrix}{1}&{2}&{-1}&2\\{0}&{\boxed{-7}}&{4}&-6\\{0}&{-7}&{4}&-6\\ 0&-2&0&1\\0&-3&4&-8\end{bmatrix}\begin{matrix}\sim \\R_3-R_2\\7R_4-2R_2\\7R_5-3R_2\end{matrix}\\ &\begin{bmatrix}{1}&{2}&{-1}&2\\{0}&{-7}&{4}&-6\\{0}&{0}&{0}&0\\ 0&0&-8&19\\0&0&16&-38\end{bmatrix}\begin{matrix}\sim \\R_3\leftrightarrow R_5 \end{matrix}\\ &\begin{bmatrix}{1}&{2}&{-1}&2\\{0}&{-7}&{4}&-6\\{0}&{0}&{\boxed{16}}&-38\\ 0&0&-8&19\\0&0&0&0\end{bmatrix}\begin{matrix}\sim \\2R_4+R_3 \end{matrix}\\ &\begin{bmatrix}{1}&{2}&{-1}&2\\{0}&{-7}&{4}&-6\\{0}&{0}&{16}&-38\\ 0&0&-8&19\\0&0&0&0\end{bmatrix}\begin{matrix}\sim \\2R_4+R_3 \end{matrix}\\ &\begin{bmatrix}{\boxed1}&{2}&{-1}&2\\{0}&{\boxed{-7}}&{4}&-6\\{0}&{0}&{\boxed{16}}&-38\\ 0&0&0&0\\0&0&0&0\end{bmatrix}\Rightarrow \text{rank }A=3. \end{aligned}$$