Question

The rank of the matrix \( A = \begin{bmatrix} 1 & 1 & 3 \\ 2 & 2 & 1 \\ 1 & 1 & 1 \end{bmatrix} \) is:

• A. 3
• B. 2
• C. 4
• D. 1

Hint:

To determine the rank of a matrix, reduce it to row echelon form and count the number of non-zero rows.

Answer 1

The Correct Option is B

We are given the matrix \( A \) as:
\[ A = \begin{bmatrix} 1 & 1 & 3 \\ 2 & 2 & 1 \\ 1 & 1 & 1 \end{bmatrix} \] To determine the rank of the matrix, we can perform row operations to reduce the matrix to its row echelon form. Step 1: Subtract 2 times the first row from the second row: \[ \begin{bmatrix} 1 & 1 & 3\\ 0 & 0 & -5 \\ 1 & 1 & 1 \end{bmatrix} \] Step 2: Subtract the first row from the third row: \[ \begin{bmatrix} 1 & 1 & 3 \\ 0 & 0 & -5 \\ 0 & 0 & -2 \end{bmatrix} \] Step 3: Multiply the second row by \( -\frac{1}{5} \) and the third row by \( -\frac{1}{2} \): \[ \begin{bmatrix} 1 & 1 & 3 \\ 0 & 0 & 1 \\ 0 & 0 & 1 \end{bmatrix} \] Step 4: Subtract the third row from the first row: \[ \begin{bmatrix} 1 & 1 & 2 \\ 0 & 0 & 1 \\ 0 & 0 & 1 \end{bmatrix} \] Now, the matrix is in row echelon form, and we can see that there are two non-zero rows. Hence, the rank of the matrix is 2. Thus, the rank of matrix \( A \) is 2.