Question

Determine the rank of the matrix:
\[ \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{pmatrix} \]

• A. 1
• B. 2
• C. 3
• D. 0

Hint:

To find the rank, reduce the matrix to row echelon form and count non-zero rows.

Answer 1

The Correct Option is A

To find the rank of the matrix $A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{pmatrix}$, we perform row reduction to echelon form.
Step 1: Apply row operations:
$R_2 \gets R_2 - 2R_1$:
\[ \begin{pmatrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 3 & 6 & 9 \end{pmatrix} \]
$R_3 \gets R_3 - 3R_1$:
\[ \begin{pmatrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \]
Step 2: The matrix has only one non-zero row.
The rank is the number of non-zero rows in the echelon form, which is 1.
Thus, the rank is 1.