Let $A$ be an $n \times n$ matrix. The set of all $n \times n$ matrices $X$ that satisfies $\left(A^{2}-3 I\right) X=O$ is not be closed under the matrix addition.
True $\square$ False
Justification:

Step 1 ：The question is asking whether the set of all \(n \times n\) matrices \(X\) that satisfies \(\left(A^{2}-3 I\right) X=O\) is closed under the matrix addition. In other words, if \(X_1\) and \(X_2\) are two matrices in the set, is their sum \(X_1 + X_2\) also in the set?

Step 2 ：To answer this question, we need to check whether \((A^2 - 3I)(X_1 + X_2) = O\). If this equation holds, then the set is closed under matrix addition. If not, then it is not closed.

Step 3 ：We can use the distributive property of matrix multiplication to simplify the left-hand side of the equation: \((A^2 - 3I)(X_1 + X_2) = A^2X_1 + A^2X_2 - 3IX_1 - 3IX_2 = (A^2X_1 - 3IX_1) + (A^2X_2 - 3IX_2)\).

Step 4 ：Since \(X_1\) and \(X_2\) are in the set, we know that \(A^2X_1 - 3IX_1 = O\) and \(A^2X_2 - 3IX_2 = O\). Therefore, \((A^2X_1 - 3IX_1) + (A^2X_2 - 3IX_2) = O + O = O\).

Step 5 ：So, the set of all \(n \times n\) matrices \(X\) that satisfies \(\left(A^{2}-3 I\right) X=O\) is closed under the matrix addition.

Step 6 ：Final Answer: \(\boxed{\text{False}}\)