Find the rank of the martrix `A=[(1,2,3),(2,4,7),(3,6,10)]` by reducing into the Echelon form.

Answer

To find the rank of the matrix \( A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 7 \\ 3 & 6 & 10 \end{pmatrix} \) by reducing it to Echelon form, we will perform row operations. Here’s the step-by-step solution:

### Step 1: Write down the matrix
We start with the matrix \( A \):
\[ A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 7 \\ 3 & 6 & 10 \end{pmatrix} \]

### Step 2: Eliminate the first column below the first row
We will use the first row to eliminate the entries below it in the first column.

- For \( R_2 \) (the second row), we perform the operation:
\[ R_2 \leftarrow R_2 - 2R_1 \]
Calculation:
\[ R_2 = \begin{pmatrix} 2 & 4 & 7 \end{pmatrix} - 2 \cdot \begin{pmatrix} 1 & 2 & 3 \end{pmatrix} = \begin{pmatrix} 2-2 & 4-4 & 7-6 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 1 \end{pmatrix} \]

- For \( R_3 \) (the third row), we perform the operation:
\[ R_3 \leftarrow R_3 - 3R_1 \]
Calculation:
\[ R_3 = \begin{pmatrix} 3 & 6 & 10 \end{pmatrix} - 3 \cdot \begin{pmatrix} 1 & 2 & 3 \end{pmatrix} = \begin{pmatrix} 3-3 & 6-6 & 10-9 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 1 \end{pmatrix} \]

After these operations, the matrix becomes:
\[ \begin{pmatrix} 1 & 2 & 3 \\ 0 & 0 & 1 \\ 0 & 0 & 1 \end{pmatrix} \]

### Step 3: Eliminate the third row
Next, we will eliminate the third row using the second row:
\[ R_3 \leftarrow R_3 - R_2 \]
Calculation:
\[ R_3 = \begin{pmatrix} 0 & 0 & 1 \end{pmatrix} - \begin{pmatrix} 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 \end{pmatrix} \]

Now the matrix is:
\[ \begin{pmatrix} 1 & 2 & 3 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix} \]

### Step 4: Identify the rank
The rank of a matrix is defined as the number of non-zero rows in its Echelon form. In our case, we have:
- Two non-zero rows: \( \begin{pmatrix} 1 & 2 & 3 \end{pmatrix} \) and \( \begin{pmatrix} 0 & 0 & 1 \end{pmatrix} \)
- One zero row: \( \begin{pmatrix} 0 & 0 & 0 \end{pmatrix} \)

Thus, the rank of the matrix \( A \) is:
\[ \text{Rank}(A) = 2 \]

### Summary
The rank of the matrix \( A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 7 \\ 3 & 6 & 10 \end{pmatrix} \) is \( 2 \).

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