GATE 2021 ST – Question 56

Matrix Theory · Vector Spaces and Linear Systems · 2 marks · Numerical answer

Let A=[a u₁ u₂ u₃],B=[b u₁ u₂ u₃],C=[u₂ u₃ u₁ a+b] be 4×4 matrices of column vectors. If det A=6,det B=2,find det(A+B)−det C.

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Correct answer: 72 to 72

Explanation

A+B=[a+b,2u₁,2u₂,2u₃],so its determinant is 2³(det A+det B)=8·8=64. Moving the first column of [a+b,u₁,u₂,u₃] to the end requires three swaps,and the resulting cyclic reordering of the other three columns is even. Hence det C=−(6+2)=−8. Difference=64−(−8)=**72**.

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