Question

Find the rank of the matrix: \[ \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 2 \\ 2 & 5 & 6 & 4 \\ 2 & 6 & 8 & 5 \end{bmatrix} \] Rank = ?

Hint:

Rank = number of linearly independent rows (or columns). Perform row reduction to echelon form to find it efficiently.

Answer 1

Correct Answer: 3

Step 1: Row operations. 
Subtract $R_1$ from $R_2$, $R_3$, and $R_4$: \[ \begin{bmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 1 \\ 1 & 4 & 5 & 3 \\ 1 & 5 & 7 & 4 \end{bmatrix} \Rightarrow \begin{bmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 1 \\ 0 & 3 & 4 & 2 \\ 0 & 4 & 6 & 3 \end{bmatrix}. \]

Step 2: Eliminate further.
Subtract $3R_2$ from $R_3$, and $4R_2$ from $R_4$: \[ \begin{bmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 1 \\ 0 & 0 & -2 & -1 \\ 0 & 0 & -2 & -1 \end{bmatrix}. \]

Step 3: Simplify.
$R_4 - R_3 $$\Rightarrow$$ 0. \text{Hence, 3 nonzero rows} $$\Rightarrow$$ \text{rank} = 3$. \[ \boxed{3.} \]