$∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} a & b & c & d a & a + b & a + b + c & a + b + c + d a & 2 a + b & 3 a + 2 b + c & 4 a + 3 b + 2 c + d a & 3 a + b & 6 a + 3 b + c & 10 a + 6 b + 3 c + d \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣$

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Solution

$∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} a & b & c & d a & a + b & a + b + c & a + b + c + d a & 2 a + b & 3 a + 2 b + c & 4 a + 3 b + 2 c + d a & 3 a + b & 6 a + 3 b + c & 10 a + 6 b + 3 c + d \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣$

$= ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} a & b & c & d 0 & a & a + b & a + b + c 0 & a & 2 a + b & 3 a + 2 b + c 0 & a & 3 a + b & 6 a + 3 b + c \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ [R_{4} \to R_{4} - R_{3}; R_{3} \to R_{3} - R_{2}; R_{2} \to R_{2} - R_{1}]$

$= a ∣ ∣ ∣ ∣ \begin{matrix} a & a + b & a + b + c a & 2 a + b & 3 a + 2 b + c a & 3 a + b & 6 a + 3 b + c \end{matrix} ∣ ∣ ∣ ∣$

$= a ∣ ∣ ∣ ∣ \begin{matrix} a & a + b & a + b + c 0 & a & 2 a + b 0 & a & 3 a + b \end{matrix} ∣ ∣ ∣ ∣ [R_{3} \to R_{3} - R_{2}; R_{2} \to R_{2} - R_{1}]$

$= a^{2} ∣ ∣ ∣ \begin{matrix} a & 2 a + b a & 3 a + b \end{matrix} ∣ ∣ ∣$

$= a^{2} (3 a^{2} + a b - 2 a^{2} - a b) = a^{4}$

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Similar questions

∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} a & b & c & d a & a + b & a + b + c & a + b + c + d a & 2 a + b & 3 a + 2 b + c & 4 a + 3 b + 2 c + d a & 3 a + b & 6 a + 3 b + c & 10 a + 6 b + 3 c + d \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣

a^{4}

a (2 a + b - c) - b (a - 2 b + 2 c)

3 c (- 2 a + 3 b + c)

∣ ∣ ∣ ∣ \begin{matrix} a & a + b & a + b + c 2 a & 3 a + 2 b & 4 a + 3 b + 2 c 3 a & 6 a + 3 b & 10 a + 6 b + 3 c \end{matrix} ∣ ∣ ∣ ∣ = a^{3}

∣ ∣ ∣ ∣ \begin{matrix} a & a + b & a + b + c 2 a & 3 a + 2 b & 4 a + 3 b + 2 c 3 a & 6 a + 3 b & 10 a + 6 b + 3 c \end{matrix} ∣ ∣ ∣ ∣

∣ ∣ ∣ ∣ \begin{matrix} a & a + b & a + b + c 2 a & 3 a + 2 b & 4 a + 3 b + 2 c 3 a & 6 a + 3 b & 10 a + 6 b + 3 c \end{matrix} ∣ ∣ ∣ ∣ = a^{3}

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