Find the nullity of the following matrix: \[ A = \begin{pmatrix} 1 & 3 & 2 \ 2 & 6 & 4\ 3 & 9 & 6 \end{pmatrix} \]

Step 1 ：Step 1: To find the nullity, we first have to find the rank of the matrix. In order to find the rank of the matrix, we perform row reductions until we get the matrix in row-echelon form.

Step 2 ：Step 2: Perform row operations: \( R2 - 2R1 \rightarrow R2 \) and \( R3 - 3R1 \rightarrow R3 \) to obtain: \[ A = \begin{pmatrix} 1 & 3 & 2 \ 0 & 0 & 0\ 0 & 0 & 0 \end{pmatrix} \]

Step 3 ：Step 3: The rank of the matrix is the number of non-zero rows, hence rank(A) = 1.

Step 4 ：Step 4: The nullity of a matrix is given by the formula: nullity(A) = n - rank(A), where n is the number of columns. Here, n = 3 and rank(A) = 1

Step 5 ：Step 5: Plug these values into the formula to find the nullity: nullity(A) = 3 - 1