I'm supposed to "use properties of determinants to evaluate the determinant by inspection" on this matrix:
$$\begin{bmatrix} 4 & 1& 3\\
-2 & 0 &-2 \\
5 & 4 & 1\end{bmatrix}.$$
I don't see anything (zero rows, ways to transform the matrix) that would make it immediately obvious what the determinant is. What am I missing?
On
If you subtract the third column from the first one, which is a valid transformation with respect to the determinant (it will leave it unchanged), you will get: $$\begin{bmatrix} 1 & 1& 3\\ 0 & 0 &-2 \\ 4 & 4 & 1\end{bmatrix}.$$
Now it's clear that the first two columns are the same, and that means that the determinant must be $0$.
Addendum: if you are wondering why the determinant must be $0$ in this case, consider what would happen if we were to repeat the same process again, subtracting the second column from the first one.
Take the third column away from the first. This leaves column 1 and 2 equal, thus the determinant = 0