Discussion
The matrix
\[ \begin{pmatrix} 1 & 1 & 1 & 1 \\ 2 & 2 & 2 & 2 \\ 1 & 3 & 3 & 3 \\ 4 & 4 & 4 & 4 \end{pmatrix} \]
has rank :
Answer: B
The rank is 2.
Row 2 is twice row 1 and row 4 is four times row 1, so they add nothing new. Row 3, \((1, 3, 3, 3)\), is not a multiple of row 1, because its first entry differs from the rest. So the row space is spanned by two independent rows:
\[ R_{3} - R_{1} = (0, 2, 2, 2) \neq 0 \]
Every \(3 \times 3\) minor vanishes, since any three rows include two proportional ones, while the minor from rows 1 and 3, columns 1 and 2, is \(1 \cdot 3 - 1 \cdot 1 = 2 \ne 0\). Rank 1 misses row 3; ranks 3 and 4 treat the proportional rows as independent.
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