Problem 28 - Discrete Mathematics with Applications

Question

Let \(\mathrm{A}, \mathrm{B},\) and \(\mathrm{C}\) 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 ). $$A+(B+C)=(A+B)+C$$

Step-by-Step Solution

Answer

To prove that \(A+(B+C) = (A+B)+C\), consider the ij-th entry of both matrices: - For \((A + B) + C\), the ij-th entry is \((a_{ij} + b_{ij}) + c_{ij}\). - For \(A + (B + C)\), the ij-th entry is \(a_{ij} + (b_{ij} + c_{ij})\). Using the associative law for real numbers, we have \((a_{ij} + b_{ij}) + c_{ij} = a_{ij} + (b_{ij} + c_{ij})\), so both matrices are equal, proving the associative law of addition for matrices.

Step 1: Define matrix addition

Matrix addition is defined element-wise, i.e., given two matrices A and B, their sum, denoted as A + B, is a matrix C, where the ij-th entry of C is given by: \[c_{ij} = a_{ij} + b_{ij}\]

Step 2: Consider (A + B) + C

Now, let's compute the elements of matrix (A + B) + C. The ij-th entry of the matrix (A + B) is: \[((A+B)_{ij} ) = a_{ij} + b_{ij}\] So, the ij-th entry of matrix (A + B) + C is: \[((A + B) + C)_{ij} = (a_{ij} + b_{ij}) + c_{ij}\]

Step 3: Consider A + (B + C)

Next, let's compute the elements of matrix A + (B + C). The ij-th entry of matrix (B + C) is: \[((B+C)_{ij} ) = b_{ij} + c_{ij}\] So, the ij-th entry of matrix A + (B + C) is: \[(A + (B + C))_{ij} = a_{ij} + (b_{ij} + c_{ij})\]

Step 4: Compare the ij-th entries of both matrices

Now, we will show that the ij-th entry of both matrices is the same: \[((A + B) + C)_{ij} = (a_{ij} + b_{ij}) + c_{ij}\] \[(A + (B + C))_{ij} = a_{ij} + (b_{ij} + c_{ij})\] Using the associative law for real numbers, we get: \[(a_{ij} + b_{ij}) + c_{ij} = a_{ij} + (b_{ij} + c_{ij})\] Since the ij-th entries of both matrices are equal, the two matrices are equal, and we have proved that: \[A+(B+C)=(A+B)+C\]

Key Concepts

Associative Property of Matrix Addition

When working with matrices, the **associative property** plays a crucial role in simplifying and solving complex problems. This property states that when adding three or more matrices, the grouping of the matrices does not affect their sum. In mathematical terms, for any matrices \(A, B,\) and \(C\) of the same size, the associative property is given by:\[A + (B + C) = (A + B) + C\]This means that it doesn't matter if you add \(B\) and \(C\) first and then add \(A\), or if you add \(A\) and \(B\) first and then add \(C\). The final result will be the same either way.
This is because the addition of individual elements obeys the associative law for real numbers, i.e., \((a + b) + c = a + (b + c)\).Some key points about associative property in matrix addition:

• Helps in rearranging matrices in a way that may simplify calculations.
• Ensures consistent results regardless of grouping.
• Applicable only when matrices have the same dimensions.

Understanding this concept allows for flexibility in computations, making problem-solving more effective when matrices are involved.

Basics of Matrix Algebra

**Matrix Algebra** involves the study of various operations that can be performed with matrices, such as addition, subtraction, multiplication, and finding determinants and inverses. It's a significant part of linear algebra with numerous applications in engineering, physics, computer science, and more.In matrix algebra, matrices are used to represent and solve linear equations, transform geometric objects, and manage data efficiently.
One of the simplest operations in matrix algebra is matrix addition, which is always performed element-wise. For two matrices \(A\) and \(B\) with the same dimensions, the sum matrix \(C = A + B\) will have entries:\[c_{ij} = a_{ij} + b_{ij}\]where \(i\) and \(j\) denote the row and column positions respectively.Other important operations in matrix algebra include:

• **Matrix Multiplication:** More complex, non-element-wise, and requires a special rule for computation.
• **Transpose of a Matrix:** Flips a matrix over its diagonal, turning its rows into columns and vice versa.
• **Inverse of a Matrix:** Only some matrices have an inverse, which can be used to solve matrix equations.

Matrix algebra provides powerful tools for solving linear systems and understanding transformations that are otherwise complex to handle.

Understanding Matrix Equality

**Matrix Equality** is simple yet fundamental in matrix operations. It establishes that two matrices are equal if and only if they have the same dimensions and every corresponding element is identical. In mathematical terms, matrices \(A\) and \(B\) are equal if:\[A = B \iff a_{ij} = b_{ij} \text{ for all } i, j\]Hence, matrix equality ensures that each entry of one matrix matches precisely with the corresponding entry of another.A few aspects of matrix equality to keep in mind:

• **Dimensions Must Match:** Both matrices must be of the same order \(m \times n\).
• **Element-wise Comparison:** Every single element in the matrices is compared, and their equality must hold true throughout.
• Used frequently in matrix proofs and operations, such as determining if two different methods of calculation yield the same result.

Understanding matrix equality is crucial for validating solutions in matrix algebra, ensuring that matrices inferred to be the same genuinely represent identical quantities or transformations.