**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} 6 & 12 & 6 \\ 1 & 2 & 1 \\ 4 & 8 & 4 \end{bmatrix}$ is a 3 x 3 matrix

$ \therefore \rho(A) \leq 3$

$ |A| = \begin{vmatrix} 6 & 12 & 6 \\ 1 & 2 & 1 \\ 4 & 8 & 4 \end{vmatrix} = 6(8-8)-12(4-4)+6(8-8) = 0$

$ \rho(A) \neq 3$

$ \therefore \rho(A) \leq 2$

All 2nd order minors are also 0.

$ \begin{vmatrix} 6 & 12 \\ 1 & 2 \end{vmatrix} = \begin{vmatrix}2 & 1 \\ 8 & 4 \end{vmatrix} = \begin{vmatrix}12 & 6 \\ 8 & 4 \end{vmatrix} = 0 $ etc..

$ \therefore \rho(A) \neq 2$

$ \rho(A) = 1$ because there are nonzero minors of order 1.

Method 2

$ A = \begin{bmatrix} 6 & 12 & 6 \\ 1 & 2 & 1 \\ 4 & 8 & 4 \end{bmatrix}\: \: R_3 \rightarrow R_3-4R_2$

$ \sim \begin{bmatrix} 6 & 12 & 6 \\ 1 & 2 & 1 \\ 0 & 0 & 0 \end{bmatrix} R_2 \rightarrow R_2 - \large\frac{1}{6}R_1$

$ \sim \begin{bmatrix} 6 & 12 & 6 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$

The last equivalent matrix is in echeron form and there is one nonzero row.

$ \therefore \rho(A) = 1$