GATE 2025 MA – Question 49
Maximize Z=3x₁+5x₂ with x₁+x₃=4, 2x₂+x₄=12, 3x₁+2x₂+x₅=18, x≥0. Optimal basis is (x₃,x₂,x₁), with inverse $B^{-1}=\begin{pmatrix}\alpha&\beta&-\beta\\0&\gamma&0\\0&-\beta&\beta\end{pmatrix}$. Let (p,q,r) be optimal dual variables. Which statements hold?
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Correct answer: (A) α+3β+2γ=3; (C) p+q+r=5/2
Explanation
Basis columns are (1,0,0)ᵀ,(0,2,2)ᵀ,(1,0,3)ᵀ. Solving yields α=1,β=1/3,γ=1/2. Thus A gives 1+1+1=3; B gives 1−1+2=2, not 1. Dual row is $c_B^TB^{-1}=(0,5,3)B^{-1}=(0,3/2,1)$. Sum=5/2, squares=13/4, not 17/4. **A,C**.