Is there a general method to find a 3×3, or 2×2 matrices, given the determinant? I want to do a project with my students when we start to study Systems of Equations. It would be interesting if the determinant is a prime number and then work backwards.
For example, I got this by playing around: if I let
$$A = \begin{bmatrix}
10 & -17\\
31 & 2
\end{bmatrix}$$
then the $\det(A)=547$, a prime number. Not sure if this would be a good mini project or not. Thanks for any help.
Best Answer
Start with the diagonalized form $A=PDP^{-1}$ where $D=\mathrm{diag}([\lambda_1, \lambda_2,\cdots,\lambda_n])$. You can then choose the diagonal terms of $D$ such that $\det(A)=\lambda_1 \lambda_2\cdots\lambda_n$. Finally you can create a similar matrix by choosing any invertable $P$.