Let a, b, c be non-real numbers satisfying th | vedclass

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(B) matrix is non-invertible if its determinant is $0$. The determinant of the given matrix is:$|A| = 1(1 - wc) - a(w - w^2c) + b(w^2 - w^2) = 1 - wc

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$28$

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(B) matrix is non-invertible if its determinant is $0$. The determinant of the given matrix is:$|A| = 1(1 - wc) - a(w - w^2c) + b(w^2 - w^2) = 1 - wc - aw + aw^2c = 0$$1 - wc - aw + a(wc)w = 0$$(1 - aw)(1 - wc) = 0$This implies $aw = 1$ or $wc = 1$.Since $x^5 = 1$,the roots are $1, w, w^2, w^3, w^4$. Given $a, b, c$ are non-real,they must be chosen from ${w, w^2, w^3, w^4}$.Case $1$: $aw = 1 \Rightarrow a = w^{-1} = w^4$. Here $a$ is fixed ($1$ choice). $b$ can be any of the $4$ non-real roots,and $c$ can be any of the $4$ non-real roots. Total = $1 	imes 4 	imes 4 = 16$.Case $2$: $wc = 1 \Rightarrow c = w^{-1} = w^4$. Here $c$ is fixed ($1$ choice). $a$ can be any of the $4$ non-real roots,and $b$ can be any of the $4$ non-real roots. Total = $4 	imes 4 	imes 1 = 16$.Intersection: $aw = 1$ and $wc = 1 \Rightarrow a = w^4$ and $c = w^4$. Here $b$ can be any of the $4$ non-real roots. Total = $1 	imes 4 	imes 1 = 4$.By Inclusion-Exclusion Principle,total matrices = $16 + 16 - 4 = 28$.

Let $a, b, c$ be non-real numbers satisfying the equation $x^5 = 1$ and $S$ be the set of all non-invertible matrices of the form $\begin{bmatrix} 1 & a & b \\ w & 1 & c \\ w^2 & w & 1 \end{bmatrix}$,where $w = e^{\frac{i 2\pi}{5}}$. Then the number of distinct matrices in the set $S$ is:

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