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

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

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

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

(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.

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