Question

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

• A. \( 1 \)
• B. \( 3 \)
• C. \( 2 \)
• D. \( 0 \)

Hint:

The rank of a matrix is the number of linearly independent rows (or columns) in the matrix.

Answer 1

The Correct Option is C

To find the rank of the matrix $A$, we can perform row operations to bring it to row-echelon form. $$A = \begin{bmatrix} 1 & 2 & 3
1 & 4 & 2
2 & 6 & 5 \end{bmatrix}$$ $R_2 = R_2 - R_1$: $$\begin{bmatrix} 1 & 2 & 3
0 & 2 & -1
2 & 6 & 5 \end{bmatrix}$$ $R_3 = R_3 - 2R_1$: $$\begin{bmatrix} 1 & 2 & 3
0 & 2 & -1
0 & 2 & -1 \end{bmatrix}$$ $R_3 = R_3 - R_2$: $$\begin{bmatrix} 1 & 2 & 3
0 & 2 & -1
0 & 0 & 0 \end{bmatrix}$$ The row-echelon form of the matrix has 2 non-zero rows. Therefore, the rank of the matrix A is 2.