Given a linear system $$ A\mathbf{x}=\mathbf{b} $$ where $$ A=\begin{bmatrix}2 & 1 & 0 \\ a^2 & 4 & 3a \\ 0 & 0 & 2 \end{bmatrix}, \quad\mathbf{x}=\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}, \quad\mathbf{b}=\begin{bmatrix}b_1\\b_2\\b_3\end{bmatrix}, $$ Questions: (1) Determine the eigenvalues of matrix $A$ in terms of $a$.
Also, consider the linear system defined by $B\mathbf{x}=\mathbf{b}$ : $$ B=\begin{bmatrix}-5 & 1 & 0 \\ a^2 & -3 & 3a \\ 0 & 0 & -5 \end{bmatrix} $$(2) By using the result, determine the values of $a$ so that a unique solution can be obtained. Hint: First determine the relationship $A$ and $B$ by subtracting the matrices.
So, I know that $A - B=7I_3$ which makes $B = -(7I_3 - A)$. I'm confused why do we have to go to eigenvalues if we can just make an augmented matrix from $B$ to $\mathbf{b}$ then just continue from there? What is the relation with subtracting $A$ and $B$ ?
