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(B) Let $x = a+b$ and $y = c+d$. The determinant can be expressed as the product of two matrices. Consider the matrix $M = \begin{bmatrix} 1 & 1 \ a

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Independent of $a, b, c$ and $d$

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(B) Let $x = a+b$ and $y = c+d$. The determinant can be expressed as the product of two matrices. Consider the matrix $M = \begin{bmatrix} 1 & 1 \ a & b \end{bmatrix} \begin{bmatrix} 1 & 1 \ c & d \end{bmatrix}$ is not directly applicable,but we can observe that the given determinant $\Delta$ is of the form of a product of two determinants. Specifically,$\Delta = \begin{vmatrix} 1 & 1 & 0 \ a & b & 0 \ 0 & 0 & 1 \end{vmatrix} \dots$ Upon performing row and column operations or factorizing the determinant,we find that the expression simplifies to $0$. Since $\Delta = 0$,the value of the determinant is $0$,which is a constant and therefore independent of $a, b, c$ and $d$.

If $a, b, c$ and $d$ are complex numbers,then the determinant $\Delta = \begin{vmatrix} 2 & a+b+c+d & ab+cd \\ a+b+c+d & 2(a+b)(c+d) & ab(c+d)+cd(a+b) \\ ab+cd & ab(c+d)+cd(a+b) & 2abcd \end{vmatrix}$ is

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