Question

If $f(r) = \begin{bmatrix} 3r & r+1 \\ 2r & r+2 \\ 3r & r+7 \end{bmatrix} \begin{bmatrix} 1 & r & r^2 \\ 1 & r^2 & r^4 \end{bmatrix}$ then $\sum_{r=1}^{2025} |f(r)|$ equals (where $|A|$ denotes determinant of $A$)

• A. 2025
• B. 0
• C. $(2025)^3
• D. $(2025)^2

Answer 1

Answer: (B)

0

Answer 2

The matrix $f(r)$ is the product of a $3 \times 2$ matrix and a $2 \times 3$ matrix. The rank of the $3 \times 2$ matrix is at most 2, and the rank of the $2 \times 3$ matrix is at most 2. Therefore, the rank of their product, $f(r)$, which is a $3 \times 3$ matrix, is at most $\min(2, 2) = 2$. A $3 \times 3$ matrix with rank less than 3 has a determinant of 0. Thus, $|f(r)| = 0$ for all $r$. The sum $\sum_{r=1}^{2025} |f(r)| = \sum_{r=1}^{2025} 0 = 0$.