Let $Ax=b$ be a linear system with matrix $A ∈ M_{4×4}$ and $\det(A)=3$

Question

Let $Ax=b$ be a linear system with matrix $A \in M_{4×4}$ and $\det (A) = 3$. If possible, give an example of matrices $A$ and $b$ such that the following are true

1. The system has exactly one solution.

2. The system has no solutions

3. The system has infinite solutions.

If any of these is not possible, explain why.

My first question about this question is, what is the best way to make up a matrix with a predefined determinant? I guess if I make up every element but one then I can solve for that element to get the determinant that I want. Is that the best way?

For the main part of the question, for (i), I think I could just come up with any $A$ that has the right $\det$ and then just any value for $b$

For (ii) and (iii), I think there is no way to write $A$ and $b$ to have either no solutions of infinite solutions since the determinant of $A$ is non-zero.

Answer 1

Consider that for a diagonal matrix $$ A=\begin{pmatrix}a_1 & & \\&\ddots&\\&&a_n\end{pmatrix} $$ holds $\det A = a_1\cdot\ldots\cdot a_n$. If you like to have a specific determinant, you can also consider the case $$ B=\begin{pmatrix}a & & & \\&1&&\\&&\ddots&\\&&&1\end{pmatrix} $$ then you get $\det B=a$.

As you wrote: Since $\det A\neq 0$, the system has exactly one solution, given by $x=A^{-1}b$.