If A = [
5 −1
1 2
] ,B = [
3 2
−1 0
], C = [
4 1
3 −2
] are given matrices then find
A(B − C) & AB − AC. Determine whether matrix multiplication is distributes over
subtraction.

Let's go through this step by step to find both \( A(B - C) \) and \( AB - AC \), then determine if matrix multiplication is distributive over subtraction.

We are given:

\[
A = \begin{bmatrix} 5 & -1 \\ 1 & 2 \end{bmatrix}, 
\quad B = \begin{bmatrix} 3 & 2 \\ -1 & 0 \end{bmatrix}, 
\quad C = \begin{bmatrix} 4 & 1 \\ 3 & -2 \end{bmatrix}
\]

### Step 1: Calculate \( B - C \)

Subtract corresponding elements of \( B \) and \( C \):

\[
B - C = \begin{bmatrix} 3 & 2 \\ -1 & 0 \end{bmatrix} - \begin{bmatrix} 4 & 1 \\ 3 & -2 \end{bmatrix}
= \begin{bmatrix} 3 - 4 & 2 - 1 \\ -1 - 3 & 0 - (-2) \end{bmatrix}
= \begin{bmatrix} -1 & 1 \\ -4 & 2 \end{bmatrix}
\]

### Step 2: Calculate \( A(B - C) \)

Now, multiply matrix \( A \) with \( B - C \):

\[
A(B - C) = \begin{bmatrix} 5 & -1 \\ 1 & 2 \end{bmatrix} \times \begin{bmatrix} -1 & 1 \\ -4 & 2 \end{bmatrix}
\]

Perform the matrix multiplication:

\[
A(B - C) = \begin{bmatrix} (5)(-1) + (-1)(-4) & (5)(1) + (-1)(2) \\ (1)(-1) + (2)(-4) & (1)(1) + (2)(2) \end{bmatrix}
= \begin{bmatrix} -5 + 4 & 5 - 2 \\ -1 - 8 & 1 + 4 \end{bmatrix}
= \begin{bmatrix} -1 & 3 \\ -9 & 5 \end{bmatrix}
\]

### Step 3: Calculate \( AB \)

Now, let's multiply \( A \) and \( B \):

\[
AB = \begin{bmatrix} 5 & -1 \\ 1 & 2 \end{bmatrix} \times \begin{bmatrix} 3 & 2 \\ -1 & 0 \end{bmatrix}
\]

Perform the matrix multiplication:

\[
AB = \begin{bmatrix} (5)(3) + (-1)(-1) & (5)(2) + (-1)(0) \\ (1)(3) + (2)(-1) & (1)(2) + (2)(0) \end{bmatrix}
= \begin{bmatrix} 15 + 1 & 10 + 0 \\ 3 - 2 & 2 + 0 \end{bmatrix}
= \begin{bmatrix} 16 & 10 \\ 1 & 2 \end{bmatrix}
\]

### Step 4: Calculate \( AC \)

Now, let's multiply \( A \) and \( C \):

\[
AC = \begin{bmatrix} 5 & -1 \\ 1 & 2 \end{bmatrix} \times \begin{bmatrix} 4 & 1 \\ 3 & -2 \end{bmatrix}
\]

Perform the matrix multiplication:

\[
AC = \begin{bmatrix} (5)(4) + (-1)(3) & (5)(1) + (-1)(-2) \\ (1)(4) + (2)(3) & (1)(1) + (2)(-2) \end{bmatrix}
= \begin{bmatrix} 20 - 3 & 5 + 2 \\ 4 + 6 & 1 - 4 \end{bmatrix}
= \begin{bmatrix} 17 & 7 \\ 10 & -3 \end{bmatrix}
\]

### Step 5: Calculate \( AB - AC \)

Now, subtract \( AC \) from \( AB \):

\[
AB - AC = \begin{bmatrix} 16 & 10 \\ 1 & 2 \end{bmatrix} - \begin{bmatrix} 17 & 7 \\ 10 & -3 \end{bmatrix}
= \begin{bmatrix} 16 - 17 & 10 - 7 \\ 1 - 10 & 2 - (-3) \end{bmatrix}
= \begin{bmatrix} -1 & 3 \\ -9 & 5 \end{bmatrix}
\]

### Step 6: Conclusion

We have:

\[
A(B - C) = \begin{bmatrix} -1 & 3 \\ -9 & 5 \end{bmatrix}
\quad \text{and} \quad AB - AC = \begin{bmatrix} -1 & 3 \\ -9 & 5 \end{bmatrix}
\]

Since \( A(B - C) = AB - AC \), matrix multiplication is **distributive** over subtraction.

Would you like more details or explanations on any step?

Here are five related questions:
1. How does matrix addition affect the properties of matrix operations?
2. What are the conditions for matrix multiplication to be possible?
3. Can matrix multiplication be commutative? Why or why not?
4. What is the geometric interpretation of matrix multiplication?
5. How does matrix distributive property relate to scalar distributive property?

**Tip:** When multiplying matrices, ensure the number of columns in the first matrix matches the number of rows in the second matrix for the operation to be valid.

If A = [5 −1; 1 2], B = [3 2; −1 0], C = [4 1; 3 −2] are given matrices then find A(B − C) & AB − AC. Determine whether matrix multiplication is distributive over subtraction.

Explore how matrix multiplication is distributive over subtraction with a step-by-step solution to A(B - C) & AB - AC. This problem involves matrix operations, including subtraction and multiplication, and is ideal for students in Grades 11-12.

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