Question

Suppose C is a 3 x 3 matrix such that det (C) = 4. Find det ( C + C)

Answer 1

Let the matrix be

$C = \begin{pmatrix} a & b & c\\ d & e & f\\ g & h & i \end{pmatrix}$ , the determinant is $\det(C) = a(ei-hf)+ b(fg-di)+c(dh-ge) .$

$\det (C+C) = \det(2C) = \begin{vmatrix} 2a & 2b & 2c\\ 2 d & 2e & 2f\\ 2g & 2h & 2i \end{vmatrix} = 2a(2e\cdot2i-2h\cdot2f)+ 2b(2f\cdot2g-2d\cdot2i)+2c(2d\cdot2h-2g\cdot2e) = 2^3\cdot\big(a(ei-hf)+ b(fg-di)+c(dh-ge)\big) = 8\cdot\det(C) = 32.$