Q.3 (a) Reduce the following matrix to normal form and find the rank of matrix \(A=\begin{bmatrix}1&1&1\\1&-1&-1\\3&1&1\end{bmatrix}\).
Best answer
\(R_2\to R_2-R_1,\ R_3\to R_3-3R_1\): \(\begin{bmatrix}1&1&1\\0&-2&-2\\0&-2&-2\end{bmatrix}\)
\(R_3\to R_3-R_2\): \(\begin{bmatrix}1&1&1\\0&-2&-2\\0&0&0\end{bmatrix}\)
\(C_2\to C_2-C_1,\ C_3\to C_3-C_1\): \(\begin{bmatrix}1&0&0\\0&-2&-2\\0&0&0\end{bmatrix}\)
\(C_3\to C_3-C_2\), then \(R_2\to-\frac12R_2\): \(\begin{bmatrix}1&0&0\\0&1&0\\0&0&0\end{bmatrix}=\begin{bmatrix}I_2&0\\0&0\end{bmatrix}\)
Normal form \(\begin{bmatrix}I_2&0\\0&0\end{bmatrix}\); rank of A = 2 (also \(|A|=0\) since columns 2 and 3 are equal).
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