Let the matrix be
C=(abcdefghi)C = \begin{pmatrix} a & b & c\\ d & e & f\\ g & h & i \end{pmatrix}C=⎝⎛adgbehcfi⎠⎞ , the determinant is det(C)=a(ei−hf)+b(fg−di)+c(dh−ge).\det(C) = a(ei-hf)+ b(fg-di)+c(dh-ge) .det(C)=a(ei−hf)+b(fg−di)+c(dh−ge).
det(C+C)=det(2C)=∣2a2b2c2d2e2f2g2h2i∣=2a(2e⋅2i−2h⋅2f)+2b(2f⋅2g−2d⋅2i)+2c(2d⋅2h−2g⋅2e)=23⋅(a(ei−hf)+b(fg−di)+c(dh−ge))=8⋅det(C)=32.\det (C+C) = \det(2C) = \begin{vmatrix} 2a & 2b & 2c\\ 2 d & 2e & 2f\\ 2g & 2h & 2i \end{vmatrix} = 2a(2e\cdot2i-2h\cdot2f)+ 2b(2f\cdot2g-2d\cdot2i)+2c(2d\cdot2h-2g\cdot2e) = 2^3\cdot\big(a(ei-hf)+ b(fg-di)+c(dh-ge)\big) = 8\cdot\det(C) = 32.det(C+C)=det(2C)=∣∣2a2d2g2b2e2h2c2f2i∣∣=2a(2e⋅2i−2h⋅2f)+2b(2f⋅2g−2d⋅2i)+2c(2d⋅2h−2g⋅2e)=23⋅(a(ei−hf)+b(fg−di)+c(dh−ge))=8⋅det(C)=32.
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