Question

The rank of the matrix \[ A= \begin{bmatrix} 1 & -1 & 0 & -2 -4 & 4 & 0 & 8 -2 & 1 & 2 & 4 \end{bmatrix} \] is:

• A. \(1\)
• B. \(0\)
• C. \(3\)
• D. \(2\)

Hint:

The rank of a matrix is the number of linearly independent rows or columns. If one row is a scalar multiple of another, it does not increase the rank.

Answer 1

The Correct Option is D

Step 1: Identify the row vectors of the matrix.
The matrix A has rows R₁ = [1, -1, 0, -2], R₂ = [-4, 4, 0, 8], and R₃ = [-2, 1, 2, 4].

Step 2: Examine linear dependence among the rows.
Notice that R₂ = -4·R₁, meaning the second row is simply a scalar multiple of the first. Therefore, R₁ and R₂ are linearly dependent.

Step 3: Test independence of the third row.
If R₃ were a multiple of R₁, there would exist some constant k such that [-2, 1, 2, 4] = k[1, -1, 0, -2]. The first entry forces k = -2, but the second entry would then require 1 = (-2)(-1) = 2, a contradiction. Hence R₃ is linearly independent of R₁.

Step 4: Conclude the rank.
With R₂ dependent on R₁ and R₃ independent of both, the matrix possesses exactly two linearly independent rows. The rank is therefore 2.

Step 5: Final conclusion.
The rank of the matrix is 2.