Problem 14 - Discrete Mathematics with Applications

Question

Let \(A\) be an \(m \times n\) matrix, \(B\) a \(p \times q\) matrix, and \(C\) an \(r \times s\) matrix. Under what conditions is each defined? Find the size of each when defined. (Note: \(A^{2}\) means \(A A .\) ) $$A+B$$

Step-by-Step Solution

Answer

For the matrix operation \(A+B\) to be defined, matrices A and B must have the same dimensions, meaning \(m=p\) and \(n=q\). When addition is defined, the resulting matrix, D, will also have the same dimensions as A and B, which is \(m \times n\).

Step 1: Check if the addition is defined

For two matrices to be added, they must have the same dimensions. In our case, matrix A and matrix B must have the same number of rows (m and p) and the same number of columns (n and q). So, the condition for A+B to be defined is: 1. m = p 2. n = q

Step 2: Size of A + B when defined

When addition is defined (m = p and n = q), the resulting matrix, let's call it D, will have the same dimensions as those of A and B. In this case, - The number of rows in D = m - The number of columns in D = n Thus, the size of the resulting matrix D=(A + B) will be m × n.

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is essential when dealing with matrices, which are simply arrays of numbers arranged in rows and columns. Each matrix is defined by its size, commonly denoted as 'm x n' where 'm' stands for the number of rows, and 'n' represents the number of columns within the matrix.

For example, if we have a matrix with 3 rows and 4 columns, its dimension is expressed as 3 x 4. These dimensions are not just numbers; they give us important information about what operations we can perform on the matrices, such as addition, subtraction, and multiplication.

When we talk about matrix addition – as in our original exercise featuring matrices A and B – it’s crucial to match their dimensions. If matrix A is an 'm x n' matrix and matrix B is a 'p x q' matrix, only when 'm = p' and 'n = q' (equivalent dimensions) can we add them to form a new matrix with the same dimensions.

Matrices Defined

Matrices are fundamental components in numerous fields such as mathematics, engineering, and computer science, defined as a rectangular arrangement of numbers in rows and columns. Each individual number in a matrix is known as an element or an entry.

Matrices can represent and solve systems of linear equations, transform geometric figures, and encode data for computer graphics, among other applications. They are typically enclosed by square or round brackets and are used to compactly write and work with multiple linear equations that involve several variables.

For instance, in the original exercise, matrices A, B and C are symbolic representations which can hold different sets of numbers depending on their application. Understanding the structure of matrices allows us to manipulate them in ways that solve real-world problems.

Matrix Operations

Matrix operations include addition, subtraction, multiplication, and more complex operations like finding the inverse or the determinant of a matrix. These operations are not all defined in the same way as they would be for scalar numbers.

When adding matrices, like A + B in the exercise, each element from one matrix is added to the corresponding element in the other matrix, and this is only possible when both matrices have the same dimensions. Subtraction operates on a similar principle but with the subtractive operation replacing addition.

Matrix multiplication, on the other hand, has a different set of rules and does not require the matrices to have the same dimensions. Instead, the number of columns in the first matrix must equal the number of rows in the second matrix.

It's important to note that matrix multiplication is not commutative, meaning that in general, A * B does not equal B * A. These operation rules are vital for anyone studying linear algebra or related mathematical fields.