The rank of the following (n+1) x (n+1) matrix, where a is a real number, is: | 1 a a^2 ... a^n | | 1 a a^2 ... a^n | | . . . ... . | | . . . ... . | | 1 a a^2 ... a^n | A. 1 B. 2 C. n D. Depends on the value of a

GATE 1995 · Engineering Mathematics · Rank of Matrix · medium

Answer: Rank = 1

1. Observe row structure: Every row is identical to R1 = [1, a, a^2, ..., a^n].
2. Apply row reduction: Each R_i - R_1 = [0, 0, 0, ..., 0]. After reduction, only R1 = [1, a, a^2, ..., a^n] is non-zero.