**Toolbox:**

- The rank of a matrix A is equal to $r$ if (i) A has atleast one minor of order r that does not vanish (ii) every minor of order $r+1$ or higher vanishes. $ \rho (A)=r.$
- By elementary transformations, a matrix can by reduced to echelon ( or triangular form ) so that (i) Every row of A with all its entries being 0 occurs below every row with non-zero entries. (ii) The first non-zero entry in each non-zero row is 1. (iii) The number of zeros before the first non-zero element in a row is less than the number of zeros in the next row. The rank of the matrix in echelon form is equal to the number of non-zero rows.

Method 1

$ A = \begin{bmatrix} 0 & 1 & 2 & 1 \\ 2 & -3 & 0 & 1 \\ 1 & 1 & -1 & 0 \end{bmatrix}$ is a $3$ x $4$ matrix. $ \rho(A) \leq 3$

A has 4 3rd order minors.

Consider the minor $\begin{vmatrix} 0 & 1 & 2 \\ 2 & -3 & 0 \\ 1 & 1 & -1 \end{vmatrix} = 0-1(-2-0)+2(2+3)$

$ = 2+10=12 \neq 0$

A has atleast one 3rd order minor $ \neq 0$

$ \therefore \rho(A) = 3$

Method 2

$ A = \begin{bmatrix} 0 & 1 & 2 & 1 \\ 2 & -3 & 0 & 1 \\ 1 & 1 & -1 & 0 \end{bmatrix} \sim \begin{bmatrix} 1 & 1 & 2 & 0 \\ 1 & -3 & 0 & 2 \\ 0 & 1 & -1 & 1 \end{bmatrix} (C_1 \leftrightarrow C_4)$

$ \sim \begin{bmatrix} 1 & 1 & 2 & 0 \\ 0 & -2 & -2 & 2 \\ 0 & 1 & -1 & 1 \end{bmatrix} R_2 \rightarrow R_2-R_1 \sim \begin{bmatrix} 1 & 1 & 2 & 0 \\ 0 & -2 & -2 & 2 \\ 0 & 0 & -4 & 4 \end{bmatrix} R_3 \rightarrow 2R_3+R_2$

The last equivalent matrix is in echelon form with 3 nonzero rows. $ \therefore \rho(A) = 3$