Discussion
The rank of the matrix \(\begin{bmatrix} -4 & 1 & -1 \\ -1 & -1 & -1 \\ 7 & -3 & 1 \end{bmatrix}\) is
Answer: B
The rank is 2.
Expand the determinant along the first row:
\[ \begin{aligned} |A| &= -4(-1 - 3) - 1(-1 + 7) - 1(3 + 7) \\ &= 16 - 6 - 10 = 0 \end{aligned} \]
So the rank is less than 3. The first two rows, (-4, 1, -1) and (-1, -1, -1), are not multiples of each other, so the rank is at least 2. Indeed the third row is a combination of the first two: -2 times the first row plus the second row gives (7, -3, 1). A 3 by 3 matrix cannot have rank 4, and rank 1 would need every row to be a multiple of one row.
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