Question

If \(A = \begin{bmatrix} -1 & 2 & 3 & -2 \\\ 2 & -5 & 1 & 2 \\\ 3 & -8 & 5 & 2 \\\ 5 & -12 & -1 & 6 \end{bmatrix}\), then the rank of the matrix A is
1. 2
2. 5
3. 4
4. 3

Answer 1

Correct Answer - Option 1 : 2

R₁ → -R₁

\(A = \begin{bmatrix} 1 & -2 & -3 & 2 \\\ 2 & -5 & 1 & 2 \\\ 3 & -8 & 5 & 2 \\\ 5 & -12 & -1 & 6 \end{bmatrix}\)

R₂ → R₂ - 2R₁; R₃ → R₃ - 3R₁; R₄ → R₄ - 5R₁

\(= \begin{vmatrix} 1 & -2 & -3 & 2 \\\ 0 & -1 & 7 & -2 \\\ 0 & -2 & 14 & -4 \\\ 0 & -2& 14& -4\end{vmatrix}\)

R₄ → R₄ - 2R₂; R₃ → R₃ - 2R₂

\(= \begin{vmatrix} 1 & -2 & -3 & 2 \\\ 0 & -1 & 7 & -2 \\\ 0 & 0 & 0 & 0 \\\ 0 & 0& 0& 0\end{vmatrix}\)

So, r[A] = 2