Final answer:
To show that B = C when AB = AC and A is invertible, we can equate the corresponding entries of B and C using the property of matrix multiplication. However, this is not true in general when A is not invertible.
Step-by-step explanation:
Let's start by assuming that AB = AC and A is invertible.
To show that B = C, we can use the property of matrix multiplication that states if two matrices are equal, then their corresponding entries are equal. Therefore, we can equate the corresponding entries of B and C.
Let's consider a specific entry of B and C, let's say entry (i, j). Since AB = AC, the (i, j) entry of AB is equal to the (i, j) entry of AC. This means that the dot product of the i-th row of A with the j-th column of B is equal to the dot product of the i-th row of A with the j-th column of C. Since A is invertible, its rows form a linearly independent set. Therefore, the only way for the dot products to be equal is if the corresponding entries of B and C are equal. Hence, B = C for all entries and matrices AB = AC if A is invertible.
However, this is not true in general when A is not invertible. If A is not invertible, there can be infinitely many solutions for B and C that satisfy AB = AC but are not equal. In this case, AB = AC does not imply B = C.