Question

In matrix algebra: Prove the basic property: A(B+C) = AB+AC.

Answer 1

To prove A(B+C) = AB+AC in matrix algebra, it's essential to recognize the commutative nature of matrix addition and the distributive property of matrix multiplication. By performing the calculations step by step and comparing, we can confirm that this property holds true for matrices just as it does for real numbers.

Step-by-step explanation:

To prove the basic property A(B+C) = AB+AC in matrix algebra, we must understand that matrix addition is commutative and matrix multiplication distributes over addition, which is known as the distributive property.

First, we acknowledge that for any matrices A, B, and C of compatible dimensions, the sum B+C is simply the addition of corresponding elements within the matrices. Once this sum is calculated, each element of B+C is then multiplied by the corresponding row from A when we perform A(B+C).

Second, when we perform AB and AC separately, we multiply each row of A with B and C respectively, then sum the results. This process follows the distributive property, which is a fundamental property of algebra that applies both to numbers and to matrices. The computation for AB+AC then, involves adding up the individual products obtained from AB and AC, which by the nature of matrix multiplication, results in the same product as A(B+C).

Lastly, by carrying out these operations step by step and comparing the results, we can establish that A(B+C) indeed yields the same matrix as AB+AC. Therefore, we have proved the distributive property for matrix multiplication over addition.

Answer 2

Certainly! To prove the basic property \(A(B+C) = AB + AC\) in matrix algebra, let's break down the expressions using matrix multiplication:

Given matrices:
- \(A\) is an \(m \times n\) matrix.
- \(B\) is an \(n \times p\) matrix.
- \(C\) is an \(n \times p\) matrix.

The product \(A(B+C)\) can be expanded as follows:

\[ A(B+C) = A \cdot B + A \cdot C \]

Now, let's go through the steps:

1. **Matrix Multiplication**:

\[ (A \cdot B)_{ij} = \sum_{k=1}^{n} A_{ik} \cdot B_{kj} \]

\[ (A \cdot C)_{ij} = \sum_{k=1}^{n} A_{ik} \cdot C_{kj} \]

2. **Summing the Products**:

\[ (A \cdot B + A \cdot C)_{ij} = (A \cdot B)_{ij} + (A \cdot C)_{ij} \]

\[ = \sum_{k=1}^{n} A_{ik} \cdot B_{kj} + \sum_{k=1}^{n} A_{ik} \cdot C_{kj} \]

3. **Factorizing and Combining Terms**:

\[ = \sum_{k=1}^{n} A_{ik} \cdot (B_{kj} + C_{kj}) \]

Now, observe that the expression \(\sum_{k=1}^{n} A_{ik} \cdot (B_{kj} + C_{kj})\) is precisely the definition of the matrix multiplication \(A(B+C)\). Therefore, we can conclude that \(A(B+C) = AB + AC\).

This demonstrates the basic property in matrix algebra.