Question

The rank of the matrix $A=\begin{bmatrix}2 & 3 & 1 & 4 \\ 0 & 1 & 2 & -1 \\ 0 & -2 & -4 & 2\end{bmatrix}$ is

• A. 2
• B. 3
• C. 1
• D. indeterminate

Answer 1

Answer: (A)

Given $A=\begin{bmatrix}2 & 3 & 1 & 4 \\ 0 & 1 & 2 & -1 \\ 0 & -2 & -4 & 2\end{bmatrix}$
Rank of $A \leq \min 3,4$
So, rank of $A \leq 3$
Now, we will reduce it to Echelon form
$R _{3} \rightarrow R _{3}+2 R _{2}$
$\begin{bmatrix}2 & 3 & 1 & 4 \\ 0 & 1 & 2 & -1 \\ 0 & 0 & 0 & 0\end{bmatrix}$
Here, row $R_{3}$ has all elements $0$ .
So, two non-zero rows.
Hence, rank$=2$