Question

Prove that for any two scalars \(g\) and \(k\) (a) \(k(A+B)=k A+k B\) \((b)(g+k) A=g A+k A\) (Note: To prove a result, you cannot use specific examples.)

Short answer

Both properties, distributivity of scalar multiplication over addition and associativity of scalar addition, are verified for matrices.

Step-by-step solution

Understand Scalar Multiplication Distributivity over Addition

We need to demonstrate that for any scalar \(k\) and matrices \(A\) and \(B\), the scalar multiplication distributed across the addition: \(k(A + B) = k A + k B\). This is known as the distributive property of scalar multiplication over matrix addition.

Express Matrices A and B Elements

Assume \(A = [a_{ij}]\) and \(B = [b_{ij}]\), where \(a_{ij}\) and \(b_{ij}\) are the elements of matrices \(A\) and \(B\), respectively. The sum of the matrices \(A + B\) is a matrix \([a_{ij} + b_{ij}]\).

Apply Scalar to Matrix Addition

When we compute \(k(A + B)\), we get the matrix where each element is multiplied by \(k\): \([k(a_{ij} + b_{ij})]\).

Distribute Scalar into Each Matrix before Summing

Rather than adding first, multiply scalar \(k\) with each matrix individually: \(k A = [k a_{ij}]\) and \(k B = [k b_{ij}]\). Therefore, \(k A + k B = [k a_{ij} + k b_{ij}]\).

Verify Element-wise Equivalence

Compare elements from Steps 3 and 4: \(k(a_{ij} + b_{ij}) = k a_{ij} + k b_{ij}\). Since both are equal element-wise for all \(i\) and \(j\), we confirm \(k(A + B) = k A + k B\).

Understand Scalar Addition Associativity

For two scalars \(g\) and \(k\), and a matrix \(A\), we need to prove that \((g + k) A = g A + k A\). This is referred to as the associative property of scalar addition in the context of scalar multiplication.

Expand (g + k) A Element-wise

Calculate \((g + k) A\), which results in the matrix \([(g + k) a_{ij}]\).

Distribute Scalars Separately

Compute \(g A = [g a_{ij}]\) and \(k A = [k a_{ij}]\). Adding these gives \(g A + k A = [g a_{ij} + k a_{ij}]\).

Element-wise Confirmation

Compare elements in Step 7 and Step 8: \((g + k) a_{ij} = g a_{ij} + k a_{ij}\). Since these are equal element-wise for any \(i\) and \(j\), it concludes that \((g + k) A = g A + k A\).

Key concepts

Distributive Property

The distributive property in the context of matrices and scalars relates to how multiplication of a scalar with matrices can be distributed over their addition. This means if you have a scalar \(k\) and two matrices \(A\) and \(B\), the scalar can be multiplied through to each matrix separately and then added together:

• \(k(A + B) = kA + kB\)

This works because we are effectively multiplying each element of the matrix sum \(A + B\) by \(k\).
Think of each element as behaving just like normal numbers do when you distribute. For instance,

• \([a_{ij} + b_{ij}]\) becomes \([k(a_{ij} + b_{ij})]\)
• which in turn equals \([ka_{ij} + kb_{ij}]\)

This proves that the operation of multiplication by a scalar can be "spread" over addition, ensuring the output is the same as performing scalar multiplication on each matrix separately, followed by their normal addition.

Associative Property

In scalar addition concerning matrices, the associative property states that for any two scalars \(g\) and \(k\), when you add them together and then multiply by a matrix \(A\), it’s the same as multiplying each scalar individually by the matrix and then adding those products:

• \((g + k)A = gA + kA\)

To break it down, consider calculating

• \((g+k)A\): each element \(a_{ij}\) in the matrix has \((g+k)\) applied, resulting in \([(g+k)a_{ij}]\)

Next, compute separately:

• \(gA = [ga_{ij}]\)
• \(kA = [ka_{ij}]\)
• and when added, \(gA + kA = [ga_{ij} + ka_{ij}]\)

Looking at the expressions, you see

• \((g + k)a_{ij} = ga_{ij} + ka_{ij}\)

Valid for each element, confirming that the associative property ensures both sides of the equation are consistent.

Matrix Addition

Matrix addition refers to the element-wise addition of two matrices of the same dimensions. In other words, each element of one matrix is added to the corresponding element of the other matrix. This is straightforward and follows a predictable pattern:

• Given matrices \(A = [a_{ij}]\) and \(B = [b_{ij}]\), their sum is \(A + B = [a_{ij} + b_{ij}]\)

Matrix addition is essential when using the distributive property with scalars, as it forms the basis for distributing scalar multiplication over matrix addition.
To ensure that addition is valid, both matrices must have identical dimensions. This means each matrix should have the same number of rows and columns. Violations of this rule often lead to errors in calculations.

Scalar Addition

When dealing with scalars in matrix operations, scalar addition involves simply adding two scalars together, creating a new scalar. This operation is elementary and straightforward:

• For scalars \(g\) and \(k\), \(g + k\) is just their sum.

In the context of matrix operations, scalar addition becomes important when applying the associative property, as it influences how scalars are applied to matrices.
For example, you apply \(g + k\) to a matrix \(A\) by treating \(g + k\) as a single scalar multiplying across each element of \(A\):

• \((g + k)A = [(g + k) a_{ij}]\)

This demonstrates how the addition of scalars can be integrated into matrix operations, maintaining structure and consistency in mathematical expressions.