The rank of the matrix $A=\begin{bmatrix}1&2&3\ 4&5&6\ 2&1&2\end{bmatrix}$ is :

Q: The rank of the matrix $A=\begin{bmatrix}1&2&3\\ 4&5&6\\ 2&1&2\end{bmatrix}$ is :

The rank of the matrix $A=\begin{bmatrix}1&2&3\\ 4&5&6\\ 2&1&2\end{bmatrix}$ is :

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Accepted Answer

To determine the rank of the matrix $ A = \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 2 & 1 & 2 \end{bmatrix} $, we need to find the maximum number of linearly independent rows or columns in the matrix. The rank of a matrix is equal to the number of non-zero rows in its row echelon form.

**Step-by-step Explanation:**

1. **Row Reduction:**
   We perform row operations to bring the matrix to its row echelon form.

Start with the original matrix:
   $$
   A = \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 2 & 1 & 2 \end{bmatrix}
   $$

- First, we can make the first element of the second row zero by replacing the second row with the second row minus 4 times the first row:
     $$
     R_2 = R_2 - 4R_1 \Rightarrow \begin{bmatrix} 4 & 5 & 6 \end{bmatrix} - 4 \times \begin{bmatrix} 1 & 2 & 3 \end{bmatrix} = \begin{bmatrix} 0 & -3 & -6 \end{bmatrix}
     $$

- Next, make the first element of the third row zero by replacing the third row with the third row minus 2 times the first row:
     $$
     R_3 = R_3 - 2R_1 \Rightarrow \begin{bmatrix} 2 & 1 & 2 \end{bmatrix} - 2 \times \begin{bmatrix} 1 & 2 & 3 \end{bmatrix} = \begin{bmatrix} 0 & -3 & -4 \end{bmatrix}
     $$

The matrix now looks like:
   $$
   \begin{bmatrix} 1 & 2 & 3 \ 0 & -3 & -6 \ 0 & -3 & -4 \end{bmatrix}
   $$

- To simplify further, replace the third row with the third row minus the second row:
     $$
     R_3 = R_3 - R_2 \Rightarrow \begin{bmatrix} 0 & -3 & -4 \end{bmatrix} - \begin{bmatrix} 0 & -3 & -6 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 2 \end{bmatrix}
     $$

The matrix in row echelon form is:
   $$
   \begin{bmatrix} 1 & 2 & 3 \ 0 & -3 & -6 \ 0 & 0 & 2 \end{bmatrix}
   $$

2. **Determine the Rank:**
   The row echelon form of the matrix has three non-zero rows, indicating that the rank of the matrix is 3.

**Justification for Option 1:**
- The rank of the matrix is 3, as there are three linearly independent rows in the row echelon form. Therefore, Option 1 is correct.

**Explanation for Incorrect Options:**
- **Option 2 (Rank 2):** This option is incorrect because the matrix has three non-zero rows in its row echelon form, not two.
- **Option 3 (Rank 1):** This option is incorrect because the matrix has more than one non-zero row in its row echelon form.
- **Option 4 (Rank 0):** This option is incorrect because the matrix is not the zero matrix; it has non-zero rows.

Thus, the correct answer is Option 1, with the rank of the matrix being 3.