GATE CS 2024 Set 2 — Question 47

Let's analyze each statement:(A) TRUE: A fundamental property of determinants is that swapping any two rows of a square matrix multiplies the determinant by $-1$. Therefore, $\det(B) = -\det(A)$.(B) TRUE: A matrix is invertible if and only if its determinant is non-zero. Since $\det(B) = -\det(A)$, if $\det(A) \neq 0$, then $\det(B) \neq 0$. Thus, if $A$ is invertible, $B$ is also invertible.(C) FALSE: Swapping rows generally destroys symmetry. For example, let $A = \begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix}$, which is symmetric. Swapping the two rows results in $B = \begin{pmatrix} 2 & 3 \\ 1 & 2 \end{pmatrix}$. Since $B_{12} = 3$ and $B_{21} = 1$, $B$ is not symmetric.(D) FALSE: The trace is the sum of the diagonal elements. Swapping rows changes which elements are on the diagonal. For example, let $A = \begin{pmatrix} 1 & 2 \\ 3 & -1 \end{pmatrix}$. Trace($A$) $= 1 + (-1) = 0$. Swapping the rows gives $B = \begin{pmatrix} 3 & -1 \\ 1 & 2 \end{pmatrix}$. Trace($B$) $= 3 + 2 = 5 \neq 0$.