Problem 45 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A\) and \(B\) are matrices of the same size and \(c\) is a scalar, then \(c(A+B)=c A+c B\).

Step-by-Step Solution

Answer

The statement \(c(A + B) = cA + cB\) is true because the scalar \(c\) distributes over the matrix addition, following the distributive property. This is shown through the calculations, where each element of the resulting matrices follows the distributive property, holding true for matrices of the same size and any scalar \(c\).

Step 1: Review matrix addition and scalar multiplication rules

Remember the rules for matrix addition and scalar multiplication. Given two matrices \(A\) and \(B\) of the same size, the sum \((A + B)\) has the same size, and each element is the sum of the corresponding elements from \(A\) and \(B\). For example, if \(A = \begin{pmatrix}a_{11} & a_{12}\\a_{21} & a_{22}\end{pmatrix}\) and \(B = \begin{pmatrix}b_{11} & b_{12}\\b_{21} & b_{22}\end{pmatrix}\), then: \((A + B) = \begin{pmatrix}a_{11} + b_{11} & a_{12} + b_{12}\\a_{21} + b_{21} & a_{22} + b_{22}\end{pmatrix}\) Scalar multiplication involves a scalar \(c\) and a matrix \(A\), where each element of \(cA\) is \(c\) times the corresponding element of \(A\). For example, with the matrix \(A\) above, we get: \(cA = \begin{pmatrix}c a_{11} & c a_{12}\\c a_{21} & c a_{22}\end{pmatrix}\)

Step 2: Test the statement with the given rules

Now, let's test the statement \((c(A + B) = cA + cB)\) using these rules. We have: \(c(A + B) = c \begin{pmatrix}a_{11} + b_{11} & a_{12} + b_{12}\\a_{21} + b_{21} & a_{22} + b_{22}\end{pmatrix} = \begin{pmatrix}c(a_{11} + b_{11}) & c(a_{12} + b_{12})\\c(a_{21} + b_{21}) & c(a_{22} + b_{22})\end{pmatrix}\) \(cA + cB = \begin{pmatrix}c a_{11} & c a_{12}\\c a_{21} & c a_{22}\end{pmatrix} + \begin{pmatrix}c b_{11} & c b_{12}\\c b_{21} & c b_{22}\end{pmatrix} = \begin{pmatrix}c a_{11} + c b_{11} & c a_{12} + c b_{12}\\c a_{21} + c b_{21} & c a_{22} + c b_{22}\end{pmatrix}\) Comparing these results, we see that: \(c(a_{11} + b_{11}) = c a_{11} + c b_{11}\) \(c(a_{12} + b_{12}) = c a_{12} + c b_{12}\) \(c(a_{21} + b_{21}) = c a_{21} + c b_{21}\) \(c(a_{22} + b_{22}) = c a_{22} + c b_{22}\) So, \(c(A + B) = cA + cB\), which means the statement is true.

Step 3: Explain why the statement is true

The statement is true because the scalar \(c\) distributes over the matrix addition, as shown in the calculations in Step 2. The rules for matrix addition and scalar multiplication are consistent with the laws of distribution, which are applicable to real numbers. Since each element of the resulting matrices follows the distributive property, the given statement \(c(A + B) = cA + cB\) holds true for matrices of the same size and any scalar \(c\).

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation in matrix algebra that allows us to add two matrices together. To perform matrix addition, both matrices must be of the same size, meaning they have the same number of rows and columns. This size requirement is crucial because it ensures that each element of one matrix has a corresponding element in the other matrix.
The process of adding matrices is relatively straightforward. Each element in the resulting matrix is the sum of the elements in the corresponding positions of the two original matrices. For example, if we have two 2x2 matrices, \( A = \begin{pmatrix}a_{11} & a_{12}\a_{21} & a_{22}\end{pmatrix} \) and \( B = \begin{pmatrix}b_{11} & b_{12}\b_{21} & b_{22}\end{pmatrix} \), then their sum would be:
\[A + B = \begin{pmatrix}a_{11} + b_{11} & a_{12} + b_{12}\a_{21} + b_{21} & a_{22} + b_{22}\end{pmatrix}\]Always keep an eye on the matrix dimensions before performing addition, as matrices of different sizes cannot be added.

Scalar Multiplication

Scalar multiplication involves multiplying every element of a matrix by a single number, known as a scalar. The scalar essentially scales the matrix, hence the name "scalar multiplication." This operation is important because it allows us to transform matrices uniformly by changing the magnitude of each element by the same factor.
Let's take a matrix \( A = \begin{pmatrix} a_{11} & a_{12}\ a_{21} & a_{22} \end{pmatrix} \) and a scalar \( c \). When we multiply the matrix by the scalar, each element of the matrix is multiplied by \( c \):
\[cA = \begin{pmatrix} ca_{11} & ca_{12}\ ca_{21} & ca_{22} \end{pmatrix}\]This operation doesn't change the size of the matrix; it only affects the numerical values of its components. Scalar multiplication is often used in combination with other matrix operations to solve algebraic equations and systems.

Distributive Property

The distributive property is a key principle that applies to both numbers and matrices. It states that multiplying a single value across a sum of values is the same as multiplying each individual value by the single value and then summing the results.
In matrix algebra, the distributive property assures that when a scalar is multiplied by the sum of two matrices, the result is the same as multiplying the scalar by each matrix separately and then adding the results. This property is crucial because it maintains consistency with how distribution works in conventional arithmetic.
Consider two matrices \( A \) and \( B \) of the same size, and a scalar \( c \). According to the distributive property, we have:
\[c(A + B) = cA + cB\]To break it down, applying the scalar \( c \) to the sum of matrices \( A \) and \( B \) is equivalent to scaling each matrix first and then adding:

• First, add matrices: \( (A + B) \)
• Next, multiply the resulting matrix by scalar \( c \)
• Or, alternatively, multiply each of the matrices by \( c \) and then add those results

The consistency of this behavior confirms that the statement \( c(A + B) = cA + cB \) holds for any matrices \( A \) and \( B \) of the same size and any scalar \( c \). This principle is an extension of the distributive law of algebra into the realm of matrices.