8. Construct a $4 \times 4$ nonzero matrix $A$ such that the vector $\left[\begin{array}{l}1 \\ 0 \\ 0 \\ 1\end{array}\right]$ is a solution for the homogeneous system where $A$ is the standard matrix.

Step 1 ：We are asked to construct a $4 \times 4$ matrix $A$ such that the vector $\left[\begin{array}{l}1 \\ 0 \\ 0 \\ 1\end{array}\right]$ is a solution for the homogeneous system where $A$ is the standard matrix.

Step 2 ：This means that when we multiply $A$ with the vector $\left[\begin{array}{l}1 \\ 0 \\ 0 \\ 1\end{array}\right]$, we should get the zero vector.

Step 3 ：To achieve this, the first and last columns of $A$ should be negatives of each other, and the middle two columns can be any real numbers.

Step 4 ：An example of such a matrix is $A = \left[\begin{array}{cccc}1 & 2 & 3 & -1 \\ 4 & 5 & 6 & -4 \\ 7 & 8 & 9 & -7 \\ 10 & 11 & 12 & -10 \end{array}\right]$

Step 5 ：Multiplying $A$ with the vector $\left[\begin{array}{l}1 \\ 0 \\ 0 \\ 1\end{array}\right]$ gives us the zero vector, confirming that our matrix $A$ is correct.

Step 6 ：Final Answer: The $4 \times 4$ matrix $A$ that satisfies the condition is \(\boxed{\begin{array}{cccc}1 & 2 & 3 & -1 \\ 4 & 5 & 6 & -4 \\ 7 & 8 & 9 & -7 \\ 10 & 11 & 12 & -10 \end{array}}\)