Suppose C is a 3 x 3 matrix such that det (C) = 4. Show that det (C+C) is equal to 32

Step-by-step explanation:Let's consider C is a matrix given by[tex]\left[\begin{array}{ccc}a&b&c\\d&e&f\\g&h&i\end{array}\right][/tex]them determinant of matrix C can be written as[tex]\begin{vmatrix}a & b & c\\ d & e & f\\  g & h & i \end{vmatrix}\ =\ 4.....(1)[/tex]Now,[tex]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}[/tex]                   [tex]=\ \begin{vmatrix}2a & 2b & 2c\\ 2d & 2e & 2f\\  2g & 2h & 2i \end{vmatrix}[/tex]                    [tex]=\ 2\times 2\times 2\times \begin{vmatrix}a & b & c\\ d & e & f\\  g & h & i \end{vmatrix}[/tex]                    [tex]=\ 8\times 4\ \ \ \ \ \ \ \         from\ eq.(1)[/tex]                     = 32      Hence, det (C+C) = 32