Evaluate Delta = begin{vmatrix} 1 & a & bc 1 | vedclass

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(B) Applying $R_{2} ightarrow R_{2} - R_{1}$ and $R_{3} ightarrow R_{3} - R_{1}$,we get: $\Delta = \begin{vmatrix} 1 & a & bc \ 0 & b-a & ca-bc \

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$(a-b)(b-c)(c-a)$

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(B) Applying $R_{2} ightarrow R_{2} - R_{1}$ and $R_{3} ightarrow R_{3} - R_{1}$,we get: $\Delta = \begin{vmatrix} 1 & a & bc \ 0 & b-a & ca-bc \ 0 & c-a & ab-bc \end{vmatrix} = \begin{vmatrix} 1 & a & bc \ 0 & b-a & -c(b-a) \ 0 & c-a & -b(c-a) \end{vmatrix}$ Taking common factors $(b-a)$ from $R_{2}$ and $(c-a)$ from $R_{3}$: $\Delta = (b-a)(c-a) \begin{vmatrix} 1 & a & bc \ 0 & 1 & -c \ 0 & 1 & -b \end{vmatrix}$ Expanding along the first column: $\Delta = (b-a)(c-a) [1 \cdot (-b - (-c))] = (b-a)(c-a)(c-b)$ Rearranging terms to match the standard form: $\Delta = -(a-b)(c-a)(-(b-c)) = (a-b)(b-c)(c-a)$

Evaluate $\Delta = \begin{vmatrix} 1 & a & bc \\ 1 & b & ca \\ 1 & c & ab \end{vmatrix}$

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