Question

Rank of the matrix \[ A= \begin{bmatrix} 1& 0& 1& 1\\ 0& 1&-3&-1\\ 3& 1& 0& 2\\ 1& 1&-2& 0 \end{bmatrix} \] is

• A. \(4\)
• B. \(3\)
• C. \(2\)
• D. \(1\)

Hint:

The rank of a matrix equals \[ \boxed{\text{the number of non-zero rows in its row-echelon form}.} \]

Answer 1

The Correct Option is C

Step 1: Reduce the matrix using elementary row operations. Starting with \[ \begin{bmatrix} 1& 0& 1& 1\\ 0& 1&-3&-1\\ 3& 1& 0& 2\\ 1& 1&-2& 0 \end{bmatrix}, \] perform \[ R_3\rightarrow R_3-3R_1,\qquad R_4\rightarrow R_4-R_1, \] to obtain \[ \begin{bmatrix} 1& 0& 1& 1\\ 0& 1&-3&-1\\ 0& 1&-3&-1\\ 0& 1&-3&-1 \end{bmatrix}. \]

Step 2: Eliminate identical rows. Now, \[ R_3\rightarrow R_3-R_2,\qquad R_4\rightarrow R_4-R_2, \] giving \[ \begin{bmatrix} 1& 0& 1& 1\\ 0& 1&-3&-1\\ 0& 0& 0& 0\\ 0& 0& 0& 0 \end{bmatrix}. \] There are exactly two non-zero rows. Hence, \[ \boxed{\operatorname{Rank}(A)=2.} \] Therefore, \[ \boxed{(C)} \] is the correct answer.