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.

We are given three matrices \( A \), \( B \), and \( C \):

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

We need to:

1. Compute \( A(B - C) \).
2. Compute \( AB - AC \).
3. Determine whether matrix multiplication is distributive over subtraction.

### Step 1: Compute \( A(B - C) \)

First, calculate \( B - C \):

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

Now compute \( A(B - C) \):

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

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 2: Compute \( AB - AC \)

Now, we calculate \( AB \) and \( AC \) separately.

#### \( AB \):

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

#### \( AC \):

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

Now compute \( AB - AC \):

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

### Step 3: Is Matrix Multiplication Distributive Over Subtraction?

We have computed:

- \( A(B - C) = \begin{bmatrix} -1 & 3 \\ -9 & 5 \end{bmatrix} \).
- \( AB - AC = \begin{bmatrix} -1 & 3 \\ -9 & 5 \end{bmatrix} \).

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

### Conclusion:

Yes, matrix multiplication is distributive over subtraction, as shown by the equality of \( A(B - C) \) and \( AB - AC \).

---

Would you like more details on any step or have any questions?

Here are 5 related questions to deepen understanding:
1. How does matrix multiplication work in general?
2. Can the distributive property hold for non-square matrices?
3. What is the difference between the associative and distributive properties in matrix operations?
4. How can we prove distributive properties for matrices symbolically?
5. What happens when matrix dimensions don't match for certain operations?

**Tip:** Always verify the dimensions of matrices before attempting addition, subtraction, or multiplication to avoid errors.

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 the distributive property of matrix multiplication over subtraction with matrices A, B, and C. This step-by-step solution shows how to calculate A(B - C) and AB - AC, demonstrating that matrix multiplication is distributive over subtraction. Perfect for students in Grades 10-12 learning about advanced matrix operations.

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