Q.3 (a) Find the rank of the matrix by reducing it to normal form \(A=\begin{bmatrix}1&-1&-2&-3\\4&1&0&2\\0&3&1&4\\0&1&0&2\end{bmatrix}\).
Best answer
\(C_2\to C_2+C_1,\ C_3\to C_3+2C_1,\ C_4\to C_4+3C_1\): \(\begin{bmatrix}1&0&0&0\\4&5&8&14\\0&3&1&4\\0&1&0&2\end{bmatrix}\)
\(R_2\to R_2-4R_1\), then \(R_2\leftrightarrow R_4\): \(\begin{bmatrix}1&0&0&0\\0&1&0&2\\0&3&1&4\\0&5&8&14\end{bmatrix}\)
\(C_4\to C_4-2C_2\): \(\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&3&1&-2\\0&5&8&4\end{bmatrix}\)
\(R_3\to R_3-3R_2,\ R_4\to R_4-5R_2\): \(\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&-2\\0&0&8&4\end{bmatrix}\)
\(C_4\to C_4+2C_3\): \(\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&8&20\end{bmatrix}\); \(R_4\to R_4-8R_3\), then \(R_4\to\frac1{20}R_4\): \(I_4\).
Normal form \(I_4\); rank of A = 4 (consistent with \(|A|=-20\ne0\)).
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