GATE 2023 ST – Question 51

Matrix Theory · Vector Spaces and Linear Systems · 2 marks · Multiple select

A real 3×3 matrix satisfies A(0,1,1)ᵀ=(4,0,0)ᵀ,A(1,0,1)ᵀ=(0,4,0)ᵀ,and A(1,1,0)ᵀ=(0,0,4)ᵀ. Which hold?

A(1,0,0)ᵀ=(2,2,−2)ᵀ
A(0,1,0)ᵀ=(2,−2,2)ᵀ
A(1,1,1)ᵀ=(2,0,2)ᵀ
A(1,2,3)ᵀ=(8,4,0)ᵀ

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Correct answer: (B) A(0,1,0)ᵀ=(2,−2,2)ᵀ; (D) A(1,2,3)ᵀ=(8,4,0)ᵀ

Explanation

Writing columns a₁,a₂,a₃ gives a₂+a₃=(4,0,0),a₁+a₃=(0,4,0),a₁+a₂=(0,0,4). Solving yields a₁=(−2,2,2),a₂=(2,−2,2),a₃=(2,2,−2). Their sum is (2,2,2),and a₁+2a₂+3a₃=(8,4,0). Thus **B,D**.

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