Here are the answers to your matrix-related questions:

1. Does the equation AB = 0 necessarily yield A = 0 or B = 0? No, the equation $AB = 0$ does not necessarily imply $A = 0$ or $B = 0$. For example, if $A$ and $B$ are non-zero matrices, it is still possible for their product to be the zero matrix, particularly in the case of singular matrices.

2. For a general matrix $A$, can we say that $AB = AC$ yields $B = C$? No, $AB = AC$ does not necessarily imply $B = C$. This is true unless $A$ is invertible. If $A$ is not invertible (e.g., it is singular), then $B$ and $C$ can be different while still satisfying $AB = AC$.

3. In general, does $AB = BA$, if $A$ and $B$ are both square? No, $AB = BA$ is not generally true even if $A$ and $B$ are square matrices. Matrix multiplication is not commutative in general.

4. Fill the gap: if $A$ is a matrix of size $m \times n$, then its transpose $A^T$ is a matrix of size ...? If $A$ is $m \times n$, then $A^T$ is of size $n \times m$.

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Relative Questions:

1. Can you give an example of two non-zero matrices $A$ and $B$ such that $AB = 0$?
2. How does the invertibility of $A$ affect the equation $AB = AC$?
3. What are the conditions under which $AB = BA$ for two matrices $A$ and $B$?
4. How is the transpose operation related to matrix multiplication?
5. If $A$ is a square matrix, does $A^T A = A A^T$ always hold?

Tip:

When working with matrix equations, always check if the involved matrices are invertible or commutative to make accurate conclusions.