Question

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

Answer 1

Step-by-step explanation:

Let's consider C is a matrix given by

$\left[\begin{array}{ccc}a&b&c\\d&e&f\\g&h&i\end{array}\right]$

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,

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

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

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

                   $=\ 8\times 4\ \ \ \ \ \ \ \ from\ eq.(1)$

                    = 32      

Hence, det (C+C) = 32