I'm currently learning about the rank and inverses of matrices, and every time I attempt to find the inverse of a matrix with a rank smaller than it's number of rows, I find I am unable. One example of this is as follows:
\begin{bmatrix}1&1&0\\1&0&1\\1&0&1\end{bmatrix}
The rank of this matrix is 2 while it has 3 rows, and it doesn't have an inverse. Have I just run into examples where this is true, or is this always true? If it is true, why?
Best Answer
The rank of a matrix is equivalent to the number of nonzero rows after Gaussian elimination. If the rank is not maximal, then there is at least one nonzero row after elimination. Do the determinant about this row, and it will be $0$, i.e., the matrix is not invertible.