GATE 2022 MA – Question 12

Linear Algebra · Vector Spaces and Linear Transformations · 1 mark · Multiple choice

P:if M∈ℝᵐˣⁿ,m>n≥2 and rank M=n, then Mx=0 has only x=0. Q:if nonzero E∈ℝⁿˣⁿ satisfies E³=0, then I+E² is singular. Choose truth values.

Show answer and explanation

Correct answer: (C) P true, Q false

Explanation

Rank-nullity gives nullity M=n−n=0, so P holds. Since E⁴=0, (I+E²)(I−E²)=I−E⁴=I. Thus I+E² has an inverse and Q fails. **C**.

GATE 2022 MA Q47 – A linear T:ℝ²→ℝ² satisfies T(1,2)=(1,0),T(2,1)=(1,1). If T⁻¹(p,q)=(x,y), which holds?
GATE 2022 MA Q56 – In ℝ³ set W₁=x:x₁+2x₂−x₃=0,W₂=x:2x₁+3x₃=0. Which hold?
GATE 2021 MA Q11 – A is real 3×4 and B real 4×3, with AB nonsingular. P:nullity A=0. Q:BA is nonsingular.…
GATE 2021 MA Q14 – R is the reduced row echelon form of a real 4×4 A, and its third column is (0,1,0,0)ᵀ.…
GATE 2021 MA Q29 – T:ℝ³→ℝ³ is linear and T(1,1,1)=(1,−1,1),T²(1,1,1)=(1,1,1),T²(1,1,2)=(1,1,1). Find rank T.
GATE 2026 MA Q11 – Let M₃(ℝ) be all real 3×3 matrices and T₃(ℝ) its upper-triangular subspace, with usual…