Question

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

Answer 1

Let's consider C is a matrix given by

$\begin{bmatrix} a & b&c \\ d&e&f\\ g&h&i \end{bmatrix}$

them determinant of matrix C can be written as

$\begin{vmatrix} a & b&c \\ d&e&f\\ g&h&i \end{vmatrix}=4.........(1)$

Now,

$det(C+C)=\begin{vmatrix} a & b&c \\ d&e&f\\ g&h&i \end{vmatrix}+\begin{vmatrix} a & b&c \\ d&e&f\\ g&h&i \end{vmatrix}=$

$=\begin{vmatrix} 2a & 2b&2c \\ 2 d&2e&2f\\ 2g&2h&2i \end{vmatrix}=2\times2\times2\times\begin{vmatrix} a & b&c \\ d&e&f\\ g&h&i \end{vmatrix}=$

$=8\times4\space$ from eq.(1)$=$

$=32$

Answer:

$det(C+C)=32$