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)

2

Answer 2

We have $A = \begin{bmatrix} 2 & 3 & 1 & 4 \\ 0 & 1 & 2 & - 1 \\ 0 & - 2 & - 4 & 2 \end{bmatrix}_{3 \times 4}$, Considering 3×3 minor

$\begin{bmatrix} 2 & 3 & 1 \\ 0 & 1 & 2 \\ 0 & - 2 & - 4 \end{bmatrix}_{3 \times 3}$its determinant is 0.

Similarly considering , $\begin{bmatrix} 2 & 3 & 4 \\ 0 & 1 & - 1 \\ 0 & - 2 & 2 \end{bmatrix}$, $\begin{bmatrix} 2 & 1 & 4 \\ 0 & 2 & - 1 \\ 0 & - 4 & 2 \end{bmatrix}$ and

$\begin{bmatrix} 3 & 1 & 4 \ 1 & 2 & - 1 \

• 2 & - 4 & 2 \end{bmatrix}$, their determinant is 0 each rank can not be 3

Then again considering a 2×2 minor, $\begin{bmatrix} 2 & 3 \\ 0 & - 2 \end{bmatrix}$, which is non zero. Thus, rank = 2