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GATE 2023 CS – Question 18

Engineering Mathematics · Linear Algebra · 1 mark · Multiple choice

Let $A=\begin{bmatrix}1&2&3&4\\4&1&2&3\\3&4&1&2\\2&3&4&1\end{bmatrix}$ and $B=\begin{bmatrix}3&4&1&2\\4&1&2&3\\1&2&3&4\\2&3&4&1\end{bmatrix}$. Let det(A) and det(B) denote the determinants of the matrices A and B, respectively. Which one of the options given below is TRUE?

  1. det(A) = det(B)
  2. det(B) = −det(A)
  3. det(A)=0
  4. det(AB) = det(A) + det(B)

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Show answer and explanation

Correct answer: (B) det(B) = −det(A)

Explanation

B is obtained from A by swapping rows 1 and 3, so det(B) = −det(A). A is a non-singular circulant matrix, so det(A) ≠ 0.