Discussion
Rank of a matrix
\[ \begin{bmatrix} 5 & 3 & -3 & -1 \\ 3 & 2 & -2 & -1 \\ 2 & -1 & 2 & 8 \end{bmatrix} \]
is
Answer: C
The rank is 3.
A 3 × 4 matrix has rank at most 3, so the question is whether the three rows are independent. Row-reduce: subtracting \(\frac{3}{5}\) of row 1 from row 2 and \(\frac{2}{5}\) of row 1 from row 3 gives
\[ \begin{bmatrix} 5 & 3 & -3 & -1 \\ 0 & 1/5 & -1/5 & -2/5 \\ 0 & -11/5 & 16/5 & 42/5 \end{bmatrix} \]
Adding 11 times row 2 to row 3 leaves \((0, 0, 1, 4)\), which is non-zero, so there are three pivots and the rank is 3.
Rank 4 is impossible for a matrix with only three rows, and ranks 1 or 2 would need a row to be a combination of the others, which the elimination rules out.
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