Problem 96 - Precalculus

Question

Determine whether the statements are true or false. \(A+B\) is defined only if \(A\) and \(B\) have the same order.

Step-by-Step Solution

Answer

True, because matrices can only be added if they have the same order.

Step 1: Understanding Matrix Order

The order of a matrix refers to its dimensions, usually given as \(m \times n\), where \(m\) is the number of rows and \(n\) is the number of columns. For two matrices to be added, they must have the same dimensions, which means their orders must be identical.

Step 2: Matrix Addition Criteria

Matrix addition is only possible when the matrices involved have the same number of rows and the same number of columns. This ensures that each element in one matrix has a corresponding element in the other matrix to combine with.

Step 3: Evaluating the Statement

Given the requirement in Step 2, the statement "\(A+B\) is defined only if \(A\) and \(B\) have the same order" is true. Without the same dimensions, matrix addition cannot be performed.

Key Concepts

Understanding Matrix Order

When discussing matrices, one important aspect to grasp is the concept of matrix order. Simply put, the order of a matrix is another term for the dimensions of a matrix. It is expressed as \(m \times n\), where \(m\) represents the number of rows and \(n\) signifies the number of columns.
Understanding this concept is crucial because the operations you can perform on matrices often depend on their order.
For instance, two matrices can only be added together if they share the same dimensions. This means both matrices must have an equal number of rows and columns. Recognizing the order immediately tells you if certain operations are viable.

Dimensions of a Matrix

The dimensions of a matrix are like its blueprint. Knowing the size of a matrix—how many rows and columns it has—can tell you a lot about what you can do with it. For example:

• A matrix with dimensions \(3 \times 2\) has 3 rows and 2 columns.
• The total number of elements in the matrix will be the product of its dimensions: in this case, 6 elements.
• When working with matrices, lining up their dimensions ensures operations like addition can take place smoothly.

By checking the dimensions first, you prevent errors and save time, ensuring that your operations are well-founded.

Basic Matrix Operations

Matrix operations encompass a variety of mathematical procedures you can perform on matrices. Some of the most common ones include matrix addition, subtraction, and scaling by a constant factor. Here's a closer look at matrix addition:

• Matrix addition involves combining two matrices of the same dimensions, where each corresponding element gets added together.
• If matrix \(A\) is \(m \times n\) and matrix \(B\) is also \(m \times n\), you can add the two matrices to produce a new matrix \(C\).
• Each element \(c_{ij}\) in the resulting matrix \(C\) is the sum of \(a_{ij}\) and \(b_{ij}\), elements from the same position in matrices \(A\) and \(B\).

It's essential to ensure that the matrices involved in addition have the same order—you can't add matrices of different sizes.
This constraint makes understanding dimensions a fundamental step before engaging in any matrix operations.