Let's analyze each statement one by one.

Statement I: AB = BC implies A = C.

This statement is false in general. To understand why, we can explore a counterexample.

Example:

Let: $A = \begin{bmatrix} 1 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 0 \\ 1 \end{bmatrix}, \quad C = \begin{bmatrix} 1 & 0 \end{bmatrix}$ Now, calculate $AB$ and $BC$: $AB = \begin{bmatrix} 1 & 0 \end{bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix} = 0$ $BC = \begin{bmatrix} 1 & 0 \end{bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix} = 0$ So, $AB = BC$, but $A \neq C$, because $A$ is a row matrix, and $C$ is also a row matrix, but they are the same in this case purely due to coincidence with the dimensions and entries. In general, the equality $AB = BC$ does not imply that $A = C$.

Statement II: AB = AC implies B = C.

This statement is false in general as well. The key reason is that if $A$ is a non-invertible (singular) matrix, then it's possible for $AB = AC$ to hold without $B = C$.

Example:

Let: $A = \begin{bmatrix} 0 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \quad C = \begin{bmatrix} 2 \\ 2 \end{bmatrix}$ Now calculate $AB$ and $AC$: $AB = \begin{bmatrix} 0 & 0 \end{bmatrix} \begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \end{bmatrix}$ $AC = \begin{bmatrix} 0 & 0 \end{bmatrix} \begin{bmatrix} 2 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \end{bmatrix}$ So, $AB = AC$, but $B \neq C$. This happens because $A$ is the zero matrix, and multiplying anything by the zero matrix results in the zero matrix, hence the equality holds but without $B$ and $C$ being equal.

Conclusion:

• Statement I is false because $AB = BC$ does not imply $A = C$.
• Statement II is false because $AB = AC$ does not imply $B = C$, especially when $A$ is singular (non-invertible).

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Would you like more details or have any questions about these concepts? Here are some related questions to explore further:

1. What is the condition for $AB = AC$ to imply $B = C$?
2. Under what conditions can $AB = BC$ imply $A = C$?
3. How does matrix invertibility affect the validity of these statements?
4. What is the concept of matrix rank and how does it relate to these properties?
5. What does it mean for a matrix to be singular or non-invertible?

Tip: A matrix is invertible if and only if its determinant is non-zero. Understanding this property can help determine conditions under which matrix equations like $AB = AC$ and $AB = BC$ hold.