Question

Use index notation to prove the distributive law for matrix multiplication, namely: \(\mathrm{A}(\mathrm{B}+\mathrm{C})=\mathrm{AB}+\mathrm{AC}.\)

Short answer

Using index notation, the distributive law for matrix multiplication is: \(\textrm{A}(\textrm{B} + \textrm{C}) = \textrm{AB} + \textrm{AC}\).

Step-by-step solution

- Express Matrix Multiplication in Index Notation

First, express the general term of the matrix product in index notation. For two matrices \(\textrm{A}\) and \(\textrm{B}\), the element at the \(i, j\) position of the product \(\textrm{AB}\) is given by: \((\textrm{AB})_{ij} = \textrm{A}_{ik}\textrm{B}_{kj}\), where we sum over the index \(k\).

- Apply the Definition to the Given Expression

Using the definition from Step 1, write down the elements of \(\textrm{A}(\textrm{B} + \textrm{C})\): \((\textrm{A}(\textrm{B} + \textrm{C}))_{ij} = \textrm{A}_{ik}(\textrm{B} + \textrm{C})_{kj}\).

- Expand the Expression Inside the Parenthesis

Distribute the addition inside: \((\textrm{B} + \textrm{C})_{kj} = \textrm{B}_{kj} + \textrm{C}_{kj}\). Then substitute back into the previous expression: \((\textrm{A}(\textrm{B} + \textrm{C}))_{ij} = \textrm{A}_{ik}(\textrm{B}_{kj} + \textrm{C}_{kj})\).

- Distribute \(A_{ik}\) Over the Addition

Use the distributive property of scalar multiplication over addition: \(\textrm{A}_{ik}(\textrm{B}_{kj} + \textrm{C}_{kj}) = \textrm{A}_{ik}\textrm{B}_{kj} + \textrm{A}_{ik}\textrm{C}_{kj}\). So, \((\textrm{A}(\textrm{B} + \textrm{C}))_{ij} = \textrm{A}_{ik}\textrm{B}_{kj} + \textrm{A}_{ik}\textrm{C}_{kj}\).

- Separate the Sums to Show the Distributive Law

Recognize that the right side of the equation is in fact the sum of the matrix products. Thus, separate the sums: \(\textrm{A}_{ik}\textrm{B}_{kj} + \textrm{A}_{ik}\textrm{C}_{kj} = (\textrm{AB})_{ij} + (\textrm{AC})_{ij}\). Therefore, \((\textrm{A}(\textrm{B} + \textrm{C}))_{ij} = (\textrm{AB})_{ij} + (\textrm{AC})_{ij}\).

- Conclude the Proof

Since the elements are equal for all \(i, j\), it follows that as matrices: \(\textrm{A}(\textrm{B} + \textrm{C}) = \textrm{AB} + \textrm{AC}\).

Key concepts

matrix multiplication

Matrix multiplication is a fundamental concept in linear algebra. It allows us to combine two matrices to form a new one. If you multiply two matrices, the resulting matrix's elements are calculated by summing the products of corresponding elements from the rows of the first matrix and columns of the second matrix. For example, if you have matrices \(\mathbf{A}\) and \(\mathbf{B}\), the element at position (i, j) of the resulting product matrix \(\mathbf{AB}\) can be written as \( (\mathbf{AB})_{ij} = A_{ik}B_{kj} \). Here, the index \(k\) represents summing over all the entries in the corresponded row and column.

index notation

Index notation is a powerful tool used in linear algebra to express detailed mathematical operations concisely. Using index notation, we can describe matrix elements in a structured form. For instance, in matrix multiplication, \( (\mathbf{AB})_{ij} = A_{ik}B_{kj} \) clearly explains that each element in the product matrix results from summing the products of corresponding indexed elements. This not only simplifies calculations but also enhances understanding by breaking down complex operations into simpler indexed forms. This notation is highly useful for proving properties, such as the distributive property in matrix algebra.

distributive property

The distributive property is a key principle in algebra that extends to matrices. It states that for any matrices \(\mathbf{A}\), \(\mathbf{B}\), and \(\mathbf{C}\), the following is true: \(\mathbf{A}(\mathbf{B} + \mathbf{C}) = \mathbf{AB} + \mathbf{AC}\). In simpler terms, multiplying a matrix by the sum of two matrices gives the same result as multiplying the matrix by each of the two matrices separately and then adding the results. When proving this using index notation, we start by writing down the combined expression and then distribute the elements to show equivalence step by step. This property assures us that matrix operations behave predictably, similar to standard arithmetic.

linear algebra proof

Proving properties such as the distributive property in linear algebra often involves a sequence of logical steps. Using index notation facilitates this process. For the distributive property proof, we begin by expressing the product in index form: \( (\mathbf{A}(\mathbf{B}+\mathbf{C}))_{ij} = A_{ik}(B_{kj} + C_{kj}) \). Next, we distribute \( A_{ik} \) over the addition within the parentheses: \( A_{ik}B_{kj} + A_{ik}C_{kj} \). Separating the sums reveals that the left side matches the sum of individual matrix products: \( (\mathbf{AB})_{ij} + (\mathbf{AC})_{ij}\). Thus, this step-by-step proof using index notation conclusively shows \( \mathbf{A}(\mathbf{B} + \mathbf{C}) = \mathbf{AB} + \mathbf{AC} \). This logical progression not only solidifies understanding but also underpins the theoretical foundations of matrix algebra.