Let A = ( a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 ) d e t A = a 1 ( a 5 ( a 9 + 2 a 3 ) − a 6 ( a 8 + 2 a 2 ) ) − a 2 ( a 4 ( a 9 + 2 a 3 ) − a 6 ( a 7 + 2 a 1 ) ) + a 3 ( a 4 ( a 8 + 2 a 2 ) − a 5 ( a 7 + 2 a 1 ) ) = a 1 ( a 5 ( a 9 ) − a 6 ( a 8 ) ) − a 2 ( a 4 ( a 9 ) − a 6 ( a 7 ) ) + a 3 ( a 4 ( a 8 ) − a 5 ( a 7 ) ) B = ( a 1 a 2 a 3 a 4 a 5 a 6 a 7 + 2 a 1 a 8 + 2 a 2 a 9 + 2 a 3 ) = a 1 ( a 5 ( a 9 + 2 a 3 ) − a 6 ( a 8 + 2 a 2 ) ) − a 2 ( a 4 ( a 9 + 2 a 3 ) − a 6 ( a 7 + 2 a 1 ) ) + a 3 ( a 4 ( a 8 + 2 a 2 ) − a 5 ( a 7 + 2 a 1 ) ) = a 1 ( a 5 ( a 9 ) − a 6 ( a 8 ) ) − a 2 ( a 4 ( a 9 ) − a 6 ( a 7 ) ) + a 3 ( a 4 ( a 8 ) − a 5 ( a 7 ) ) + 2 ( a 1 a 3 a 5 − a 1 a 2 a 6 − a 2 a 3 a 4 + a 1 a 2 a 6 + a 2 a 3 a 4 − a 1 a 3 a 5 ) = a 1 ( a 5 ( a 9 ) − a 6 ( a 8 ) ) − a 2 ( a 4 ( a 9 ) − a 6 ( a 7 ) ) + a 3 ( a 4 ( a 8 ) − a 5 ( a 7 ) ) + 0 = a 1 ( a 5 ( a 9 ) − a 6 ( a 8 ) ) − a 2 ( a 4 ( a 9 ) − a 6 ( a 7 ) ) + a 3 ( a 4 ( a 8 ) − a 5 ( a 7 ) ) = d e t A \textsf{Let}\hspace{0.1cm}A = \begin{pmatrix}
a_1 & a_2 & a_3\\
a_4 & a_5 & a_6\\
a_7 & a_8 & a_9
\end{pmatrix}\\
\mathrm{det} A = a_1(a_5(a_9 + 2a_3) - a_6(a_8 + 2a_2)) - a_2(a_4(a_9 + 2a_3) - a_6(a_7 + 2a_1)) + a_3(a_4(a_8 + 2a_2) - a_5(a_7 + 2a_1)) = a_1(a_5(a_9) - a_6(a_8)) - a_2(a_4(a_9) - a_6(a_7)) + a_3(a_4(a_8) - a_5(a_7))\\
B =
\begin{pmatrix}
a_1 & a_2 & a_3\\
a_4 & a_5 & a_6\\
a_7 + 2a_1 & a_8 + 2a_2 & a_9 + 2a_3
\end{pmatrix} = \\a_1(a_5(a_9 + 2a_3) - a_6(a_8 + 2a_2)) - a_2(a_4(a_9 + 2a_3) - a_6(a_7 + 2a_1)) + a_3(a_4(a_8 + 2a_2) - a_5(a_7 + 2a_1)) \\= a_1(a_5(a_9) - a_6(a_8)) - a_2(a_4(a_9) - a_6(a_7)) + a_3(a_4(a_8) - a_5(a_7)) + 2(a_1a_3a_5 - a_1a_2a_6 - a_2a_3a_4 + a_1a_2a_6 + a_2a_3a_4 - a_1a_3a_5) \\= a_1(a_5(a_9) - a_6(a_8)) - a_2(a_4(a_9) - a_6(a_7)) + a_3(a_4(a_8) - a_5(a_7)) + 0 = a_1(a_5(a_9) - a_6(a_8)) - a_2(a_4(a_9) - a_6(a_7)) + a_3(a_4(a_8) - a_5(a_7)) = \mathrm{det} A Let A = ⎝ ⎛ a 1 a 4 a 7 a 2 a 5 a 8 a 3 a 6 a 9 ⎠ ⎞ det A = a 1 ( a 5 ( a 9 + 2 a 3 ) − a 6 ( a 8 + 2 a 2 )) − a 2 ( a 4 ( a 9 + 2 a 3 ) − a 6 ( a 7 + 2 a 1 )) + a 3 ( a 4 ( a 8 + 2 a 2 ) − a 5 ( a 7 + 2 a 1 )) = a 1 ( a 5 ( a 9 ) − a 6 ( a 8 )) − a 2 ( a 4 ( a 9 ) − a 6 ( a 7 )) + a 3 ( a 4 ( a 8 ) − a 5 ( a 7 )) B = ⎝ ⎛ a 1 a 4 a 7 + 2 a 1 a 2 a 5 a 8 + 2 a 2 a 3 a 6 a 9 + 2 a 3 ⎠ ⎞ = a 1 ( a 5 ( a 9 + 2 a 3 ) − a 6 ( a 8 + 2 a 2 )) − a 2 ( a 4 ( a 9 + 2 a 3 ) − a 6 ( a 7 + 2 a 1 )) + a 3 ( a 4 ( a 8 + 2 a 2 ) − a 5 ( a 7 + 2 a 1 )) = a 1 ( a 5 ( a 9 ) − a 6 ( a 8 )) − a 2 ( a 4 ( a 9 ) − a 6 ( a 7 )) + a 3 ( a 4 ( a 8 ) − a 5 ( a 7 )) + 2 ( a 1 a 3 a 5 − a 1 a 2 a 6 − a 2 a 3 a 4 + a 1 a 2 a 6 + a 2 a 3 a 4 − a 1 a 3 a 5 ) = a 1 ( a 5 ( a 9 ) − a 6 ( a 8 )) − a 2 ( a 4 ( a 9 ) − a 6 ( a 7 )) + a 3 ( a 4 ( a 8 ) − a 5 ( a 7 )) + 0 = a 1 ( a 5 ( a 9 ) − a 6 ( a 8 )) − a 2 ( a 4 ( a 9 ) − a 6 ( a 7 )) + a 3 ( a 4 ( a 8 ) − a 5 ( a 7 )) = det A
#339914 on Dec 2023
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