Problem 29 - 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 ). $$A+O=A=O+A$$

Step-by-Step Solution

Answer

To prove the theorem \(A + O = A = O + A\), we follow these steps: 1. Define the addition of matrices: Adding two matrices with the same dimensions involves adding the corresponding elements from each matrix. 2. Add matrices A and O: As all elements of matrix O are zeros, adding any of them to the corresponding elements of matrix A results in an unchanged matrix A. 3. Simplify the addition: Obtain the same result for both \(A + O\) and \(O + A\), which is matrix A. 4. Conclusion: Since we have shown that \(A + O = A\) and \(O + A = A\), we conclude that \(A + O = A = O + A\). This proves that adding a zero matrix to any other matrix results in the same matrix.

Step 1: Define the addition of matrices

Recall that the addition of two matrices with the same dimensions can be performed by adding the corresponding elements from each matrix. Let two matrices A and O be given as: A = \(\begin{bmatrix} a_{11} & a_{12} & ... & a_{1n} \\ a_{21} & a_{22} & ... & a_{2n} \\ ... & ... & ... & ... \\ a_{m1} & a_{m2} & ... & a_{mn} \end{bmatrix}\) and O = \(\begin{bmatrix} o_{11} & o_{12} & ... & o_{1n} \\ o_{21} & o_{22} & ... & o_{2n} \\ ... & ... & ... & ... \\ o_{m1} & o_{m2} & ... & o_{mn} \end{bmatrix}\) where all elements of O are zeros.

Step 2: Add matrices A and O

To add A and O, we take the corresponding elements from each matrix and add them up. The resulting matrix will be: A + O = \(\begin{bmatrix} a_{11} + o_{11} & a_{12} + o_{12} & ... & a_{1n} + o_{1n} \\ a_{21} + o_{21} & a_{22} + o_{22} & ... & a_{2n} + o_{2n} \\ ... & ... & ... & ... \\ a_{m1} + o_{m1} & a_{m2} + o_{m2} & ... & a_{mn} + o_{mn} \end{bmatrix}\)

Step 3: Simplify the addition

As all elements of O are zeros, adding each of them to the corresponding elements of A keeps A unchanged: A + O = \(\begin{bmatrix} a_{11} & a_{12} & ... & a_{1n} \\ a_{21} & a_{22} & ... & a_{2n} \\ ... & ... & ... & ... \\ a_{m1} & a_{m2} & ... & a_{mn} \end{bmatrix}\) = A We get the same result if we start adding O and A: O + A = \(\begin{bmatrix} o_{11} + a_{11} & o_{12} + a_{12} & ... & o_{1n} + a_{1n} \\ o_{21} + a_{21} & o_{22} + a_{22} & ... & o_{2n} + a_{2n} \\ ... & ... & ... & ... \\ o_{m1} + a_{m1} & o_{m2} + a_{m2} & ... & o_{mn} + a_{mn} \end{bmatrix}\) = A

Step 4: Conclusion

Since we have shown that A + O = A and O + A = A, we conclude that A + O = A = O + A. This proves that adding a zero matrix to any other matrix results in the same matrix.

Key Concepts

Zero Matrix

A zero matrix, often denoted by the letter "O," is a special type of matrix where all elements are zero. It acts as an additive identity in matrix addition. This means that when a zero matrix is added to any other matrix of the same dimensions, the original matrix remains unchanged. For example, if we have a matrix \[O = \begin{bmatrix}0 & 0 & 0\0 & 0 & 0\end{bmatrix},\]adding it to another matrix \[A = \begin{bmatrix}1 & 2 & 3\4 & 5 & 6\end{bmatrix},\]results in the same matrix A. This property can be shown mathematically as: \[A + O = A.\]Therefore, the zero matrix is an important concept in matrix operations, enhancing the understanding of identity elements.

Matrix Dimensions

Matrix dimensions define the size of a matrix and are expressed as rows by columns, written as \(m \times n\). In this notation, \(m\) represents the number of rows, while \(n\) represents the number of columns. For instance, a matrix with 3 rows and 2 columns would have dimensions \(3 \times 2\). Understanding matrix dimensions is crucial because operations like addition and multiplication require matrices to have specific dimensions to be feasible. In addition, two matrices can only be added if they are of the same dimensions.

• An example:
  • Matrix A: \(2 \times 3\)
  • Matrix B: \(2 \times 3\)
  • They can be added because their dimensions match.

Matrix Elements

Matrix elements are the individual values or numbers that compose a matrix. Each element is located at a specific position within the matrix, identified by its row and column number. Elements are usually denoted using subscripts; for example, \(a_{ij}\) refers to the element in the \(i\)-th row and the \(j\)-th column of the matrix A. Understanding elements is vital for matrix operations, as these values are used when adding, subtracting, or multiplying matrices.

• In operations:
  • For addition, corresponding elements from two matrices are summed.
  • For multiplication, elements are multiplied across rows and columns.

Hence, grasping the concept of matrix elements helps simplify complex calculations by focusing on the individual components.

Associative Property of Matrices

The associative property is a foundational concept in mathematics that also applies to matrices. It states that the way in which matrices are grouped during addition or multiplication does not change the result. In other words, if matrices A, B, and C are added together, their grouping can be altered without affecting the final sum: \[(A + B) + C = A + (B + C).\]This property is particularly useful when dealing with large matrices or complicated expressions, as it allows the rearrangement and simplification of expressions to facilitate easier computation. Ensuring correctness in matrix arithmetic, the associative property aids in verifying results and performing complex calculations effectively. Understanding this property is essential for students in order to confidently engage with matrix operations in mathematics.
Consequently, having a firm grasp on associative calculations enables more flexible handling of matrices in various scenarios.