Question

The rank of the matrix \( A = \begin{bmatrix} 1 & 2 & 3 & 0 \\ 0 & -4 & -8 & 3 \\ 2 & 4 & 3 & 2 \\ 0 & -4 & -11 & 5 \end{bmatrix} \) is:

• A. 4
• B. 2
• C. 1
• D. 3

Hint:

To find the rank of a matrix, perform row reduction (Gaussian elimination) to row echelon form. The number of non-zero rows gives the rank.

Answer 1

The Correct Option is D

We aim to compute the rank of matrix \( A \), which is the maximum number of linearly independent rows (or columns). Start with: \[ A = \begin{bmatrix} 1 & 2 & 3 & 0 \\ 0 & -4 & -8 & 3 \\ 2 & 4 & 3 & 2 \\ 0 & -4 & -11 & 5 \end{bmatrix} \] Step 1: Apply row operations to simplify. Let \( R_3 \leftarrow R_3 - 2R_1 \) \[ R_3 = [2, 4, 3, 2] - 2 \cdot [1, 2, 3, 0] = [0, 0, -3, 2] \] Matrix becomes: \[ \begin{bmatrix} 1 & 2 & 3 & 0 \\ 0 & -4 & -8 & 3 \\ 0 & 0 & -3 & 2 \\ 0 & -4 & -11 & 5 \end{bmatrix} \] Step 2: Eliminate the second column below \( R_2 \) Let \( R_4 \leftarrow R_4 - R_2 \) \[ R_4 = [0, -4, -11, 5] - [0, -4, -8, 3] = [0, 0, -3, 2] \] Now: \[ \begin{bmatrix} 1 & 2 & 3 & 0 \\ 0 & -4 & -8 & 3 \\ 0 & 0 & -3 & 2 \\ 0 & 0 & -3 & 2 \end{bmatrix} \] Step 3: Eliminate \( R_4 \leftarrow R_4 - R_3 \) \[ R_4 = [0, 0, -3, 2] - [0, 0, -3, 2] = [0, 0, 0, 0] \] Final matrix: \[ \begin{bmatrix} 1 & 2 & 3 & 0 \\ 0 & -4 & -8 & 3 \\ 0 & 0 & -3 & 2 \\ 0 & 0 & 0 & 0 \end{bmatrix} \]

This matrix has 3 non-zero rows
Rank = 3.