Question

Find the rank of the following matrix \( A \).

\[ A=\left[\begin{array}{cccc} 1 & -1 & 2 & -3 \\ 4 & 1 & 0 & 2 \\ 0 & 3 & 0 & 4 \\ 0 & 1 & 0 & 2 \end{array}\right] \]

Answer 1

\(A=\begin{bmatrix}1&-1&2&-3\\4&1&0&2\\0&3&0&4\\0&1&0&2\end{bmatrix}\)

↔ \(\begin{bmatrix}2&1&-1&-3\\0&4&1&2\\0&0&3&4\\0&0&1&2\end{bmatrix}\)

(Applying C₁ ↔ C₃ Then C₂ ↔ C₃)

↔\(\begin{bmatrix}2&1&-1&-3\\0&4&1&2\\0&0&3&4\\0&0&0&2\end{bmatrix}\) (Applying R₄ → 3R₄ - R₃)

∴ Rank of matrix A is 4.