GATE 2025 ST – Question 27

Matrix Theory · Vector Spaces and Linear Systems · 1 mark · Numerical answer

T:ℝ³→ℝ³ sends (x₁,x₂,x₃) to (3x₁+5x₂+x₃,x₃,2x₁+2x₃). Find rank T (integer).

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Correct answer: 3 to 3

Explanation

Its matrix is [[3,5,1],[0,0,1],[2,0,2]]. Expansion along the second row gives determinant −det[[3,5],[2,0]]=10≠0. Hence it is invertible and **rank=3**.

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