Problem 32 - Discrete Mathematics with Applications

Question

Let \(A, B,\) and \(C\) be any \(m \times n\) matrices, \(O\) the \(m \times n\) zero matrix, and \(c\) and \(d\) any real numbers. Prove each (see Theorem 3.12). $$(c d) A=c(d A)$$

Step-by-Step Solution

Answer

To prove the given expression \((c d) A=c(d A)\), we applied the definition of scalar multiplication to both sides of the equation. We found that the left side of the equation becomes: \((cd)A = \begin{pmatrix} (cd)a_{11} & (cd)a_{12} & \dots & (cd)a_{1n} \\ (cd)a_{21} & (cd)a_{22} & \dots & (cd)a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ (cd)a_{m1} & (cd)a_{m2} & \dots & (cd)a_{mn} \end{pmatrix}\) And the right side of the equation becomes: \(c(dA) = \begin{pmatrix} (cd)a_{11} & (cd)a_{12} & \dots & (cd)a_{1n} \\ (cd)a_{21} & (cd)a_{22} & \dots & (cd)a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ (cd)a_{m1} & (cd)a_{m2} & \dots & (cd)a_{mn} \end{pmatrix}\) Since the matrices on both sides of the equation have the same elements, we can conclude that \((cd)A = c(dA)\).

Step 1: Apply the definition of scalar multiplication

First, we need to recall the definition of scalar multiplication. When we multiply a scalar with a matrix, every element of the matrix is multiplied by the scalar. So, we can write the left side of the equation as: \((cd)A = \begin{pmatrix} (cd)a_{11} & (cd)a_{12} & \dots & (cd)a_{1n} \\ (cd)a_{21} & (cd)a_{22} & \dots & (cd)a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ (cd)a_{m1} & (cd)a_{m2} & \dots & (cd)a_{mn} \end{pmatrix}\)

Step 2: Apply the definition of scalar multiplication to the right side of the equation

Now, we need to apply the definition of scalar multiplication to the right side of the equation, dA. We get: \(dA = \begin{pmatrix} d a_{11} & d a_{12} & \dots & d a_{1n} \\ d a_{21} & d a_{22} & \dots & d a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ d a_{m1} & d a_{m2} & \dots & d a_{mn} \end{pmatrix}\) Now, we can multiply matrix dA by the scalar c: \(c (dA) = c \begin{pmatrix} d a_{11} & d a_{12} & \dots & d a_{1n} \\ d a_{21} & d a_{22} & \dots & d a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ d a_{m1} & d a_{m2} & \dots & d a_{mn} \end{pmatrix} = \begin{pmatrix} (cd)a_{11} & (cd)a_{12} & \dots & (cd)a_{1n} \\ (cd)a_{21} & (cd)a_{22} & \dots & (cd)a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ (cd)a_{m1} & (cd)a_{m2} & \dots & (cd)a_{mn} \end{pmatrix}\)

Step 3: Compare the two sides of the equation element-wise

We can now compare the elements of the matrices on both sides of the equation: \((cd)A = c(dA) \Rightarrow \begin{pmatrix} (cd)a_{11} & (cd)a_{12} & \dots & (cd)a_{1n} \\ (cd)a_{21} & (cd)a_{22} & \dots & (cd)a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ (cd)a_{m1} & (cd)a_{m2} & \dots & (cd)a_{mn} \end{pmatrix} = \begin{pmatrix} (cd)a_{11} & (cd)a_{12} & \dots & (cd)a_{1n} \\ (cd)a_{21} & (cd)a_{22} & \dots & (cd)a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ (cd)a_{m1} & (cd)a_{m2} & \dots & (cd)a_{mn} \end{pmatrix}\) Because the matrices on both sides of the equation have the same elements, we can conclude that (cd)A = c(dA).

Key Concepts

Properties of Matrices

Matrices are fundamental in mathematics and have several interesting properties that make them useful in various applications. Matrix properties include aspects like matrix addition, scalar multiplication, and matrix multiplication.
One valuable property of matrices is the **commutative property** for addition: if you have matrices \( A \) and \( B \) of the same size, then \( A + B = B + A \). However, this doesn't hold for multiplication; generally, \( AB eq BA \). This is because, in matrices, the order of multiplication matters.

Another important property is **associative property**. For multiplication, this means that if you explore three matrices, \( A \), \( B \), and \( C \), then \( (AB)C = A(BC) \). For addition, it means \( (A + B) + C = A + (B + C) \).
**Distributive properties** are also noted here, like \( A(B + C) = AB + AC \) and \( (A + B)C = AC + BC \). These properties help in breaking down complex matrix operations into smaller, manageable pieces.

Matrix Algebra

Matrix algebra involves a set of rules that allow us to add, subtract, and multiply matrices by scalars or other matrices. Scalar multiplication is one of the main operations in matrix algebra.
This operation involves multiplying each entry in a matrix by a scalar (a constant number). For instance, if you multiply a matrix \( A \) by a scalar \( c \), you effectively scale the matrix without altering its dimensions.

• **Scalar multiplication** aids in modifying matrix values while preserving the matrix structure.
• It’s important to remember that the result is still a matrix of the same size.
• This operation is distributive over matrix addition: \( c(A + B) = cA + cB \).

The matrix scalar multiplication property simplifies computations in larger matrix operations. This is pivotal for algorithms in graphics, statistics, and physical simulations, where matrices represent data sets, transformations, or complex equations.

Matrix Multiplication

Matrix multiplication is a key concept in linear algebra with its own set of rules and requirements. It differs significantly from simple arithmetic multiplication.
Key points to understand in matrix multiplication include:

• **Conformability**: Matrix \( A \) can be multiplied by matrix \( B \) if the number of columns in \( A \) is equal to the number of rows in \( B \).
• The result is a new matrix where the number of rows is from \( A \) and the number of columns is from \( B \).
• **Element calculation**: Each element of the resulting matrix is obtained by taking the dot product of the corresponding row of the first matrix and the column of the second matrix.
• This operation is **not commutative**; generally, \( AB eq BA \).

Matrix multiplication can be tricky, but it is vital in various applications like computer graphics and solving systems of equations. Understanding these foundational rules will help you manipulate matrices effectively for a wide array of mathematical and scientific calculations.