To verify that $AB = AC$ while $B \neq C$, we will compute $AB$, $AC$, and check the equality $AB = AC$. Then, we compare $B$ and $C$ to ensure they are not equal.

Given:

• $A = \begin{bmatrix} 2 & -3 \\ -4 & 6 \end{bmatrix}$,
• $B = \begin{bmatrix} 8 & 4 \\ 5 & 5 \end{bmatrix}$,
• $C = \begin{bmatrix} 5 & -2 \\ 3 & 1 \end{bmatrix}$.

Step 1: Compute $AB$

$AB = \begin{bmatrix} 2 & -3 \\ -4 & 6 \end{bmatrix} \begin{bmatrix} 8 & 4 \\ 5 & 5 \end{bmatrix}$ The resulting entries for $AB$ are computed as follows: $\text{Row 1, Col 1: } (2)(8) + (-3)(5) = 16 - 15 = 1$ $\text{Row 1, Col 2: } (2)(4) + (-3)(5) = 8 - 15 = -7$ $\text{Row 2, Col 1: } (-4)(8) + (6)(5) = -32 + 30 = -2$ $\text{Row 2, Col 2: } (-4)(4) + (6)(5) = -16 + 30 = 14$ Thus, $AB = \begin{bmatrix} 1 & -7 \\ -2 & 14 \end{bmatrix}$

Step 2: Compute $AC$

$AC = \begin{bmatrix} 2 & -3 \\ -4 & 6 \end{bmatrix} \begin{bmatrix} 5 & -2 \\ 3 & 1 \end{bmatrix}$ The resulting entries for $AC$ are computed as follows: $\text{Row 1, Col 1: } (2)(5) + (-3)(3) = 10 - 9 = 1$ $\text{Row 1, Col 2: } (2)(-2) + (-3)(1) = -4 - 3 = -7$ $\text{Row 2, Col 1: } (-4)(5) + (6)(3) = -20 + 18 = -2$ $\text{Row 2, Col 2: } (-4)(-2) + (6)(1) = 8 + 6 = 14$ Thus, $AC = \begin{bmatrix} 1 & -7 \\ -2 & 14 \end{bmatrix}$

Step 3: Verify $AB = AC$

From the calculations: $AB = \begin{bmatrix} 1 & -7 \\ -2 & 14 \end{bmatrix}, \quad AC = \begin{bmatrix} 1 & -7 \\ -2 & 14 \end{bmatrix}$ Since the matrices are identical, $AB = AC$.

Step 4: Verify $B \neq C$

$B = \begin{bmatrix} 8 & 4 \\ 5 & 5 \end{bmatrix}, \quad C = \begin{bmatrix} 5 & -2 \\ 3 & 1 \end{bmatrix}$ Clearly, $B \neq C$, as their corresponding entries are not equal.

Conclusion:

We have verified that $AB = AC$ and yet $B \neq C$.

Do you want a detailed breakdown of any part? Let me know!

---

5 Related Questions:

1. What property of matrix multiplication explains why $AB = AC$ even when $B \neq C$?
2. Can you generalize conditions under which $AB = AC$ but $B \neq C$?
3. How does the determinant of a matrix like $A$ influence such results?
4. If $A$ is invertible, would $AB = AC$ still imply $B = C$? Why or why not?
5. What happens when we extend this analysis to non-square matrices?

Tip:

Always check the properties of the left-hand matrix $A$, such as singularity or rank, when dealing with equations involving matrix products!