Discussion
The rank of the matrix \(\begin{bmatrix} 1 & 1 & 1 \\ 1 & -1 & -1 \\ 3 & 1 & 1 \end{bmatrix}\) is
Answer: A
The rank is 2. The third row is twice the first row plus the second row, so the rows are linearly dependent, while the first two rows are not multiples of each other.
\[ 2(1, 1, 1) + (1, -1, -1) = (3, 1, 1) \]Hence only two independent rows remain and the rank is not 3. It is also not 1, since the first two rows are independent, and the rank can be determined, so the other options are wrong.
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