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

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

`=\ 8* 4\ \ \ \ \ \ \ \         from\ eq.(1)`

= 32

Hence, det (C+C) = 32