Find the determinants of the following matrices

Question

Let $$\det\begin{bmatrix} a& b & c \\ d & e & f \\ g & h & i \end{bmatrix} = 5.$$ Find the determinants of the following: $$1)\quad \begin{bmatrix} a& b & c \\ g & h & i \\ d & e & f \end{bmatrix}, \quad 2)\quad \begin{bmatrix} d& e & f \\ g & h & i \\ a & b & c \end{bmatrix},\quad 3)\quad \begin{bmatrix} a& b & b \\ d & e & e \\ g & h & h \end{bmatrix}.$$

Answer 1

Hint: you have probably learned that the value of a determinant changes in certain predictable ways when you swap two rows, permute the rows, etc.

For part #3, if you can't find any clever approach, just compute the determinant from the definition: $$ \text{determinant} = a(eh-he) - b(dh - ge) + \ldots $$