Square matrix whose determinant is already known

Question

I was reading elementary algebra and suddenly I got stuck in a question which was

Write a $3*3$ square matrix $A$ whose determinant is $75$ ?

Now this can be done using hit and trial method or simply by using common sense. I'm wondering whether there is a standard method to solve this problem. I'm trying a lot but couldn't find a general/standard method to solve this. But I'm sure that there must be a method for such problems.

Answer 1

Hint:

1. What's the determinant of $\lambda\cdot A$, where $\lambda\in\mathbb R$ and $\det A$ is known?
2. Do you know of any matrix with a simple-to-calculate determinant?

Answer 2

How about :$$\left(\begin{matrix} 75 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}\right)$$ Or any other diagonal matrix where the product of the diagonal equals $75$. So the general method to find a matrix $A$ such that $det(A)= b$ is just to create a diagonal matrix where one entry on the diagonal is $b$ and the others are $1$.