Problem 8 - Advanced Engineering Mathematics

Question

In Problems 7-10, determine whether the given matrices are equal. $$ \left(\begin{array}{ll} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{ll} 1 & 0 \\ 2 & 1 \end{array}\right) $$

Step-by-Step Solution

Answer

The matrices are not equal because their elements differ.

Step 1: Understanding Matrix Equality

Two matrices are equal if and only if they have the same dimensions and all their corresponding elements are equal. In this problem, we have two 2x2 matrices. We need to compare each corresponding element of these matrices to determine if they are equal.

Step 2: Comparing Elements

Compare the individual elements from both matrices: 1. First element at position (1,1): In the first matrix, it is 1; in the second matrix, it is 1. They are equal. 2. Second element at position (1,2): In the first matrix, it is 2; in the second matrix, it is 0. They are not equal. Since there is at least one pair of corresponding elements that differ, the matrices are not equal.

Key Concepts

Matrix comparison

When we talk about matrix comparison, we're essentially discussing how we determine if two matrices are the same. This comparison is not about visual patterns but analytic equivalence. Two matrices are considered equal if both the following conditions are met:

• They have the same dimensions, meaning they have an equal number of rows and columns.
• Every corresponding element in both matrices is equal. This means that for matrices to be equal, the element at each position in the first matrix must have the same value as the element in the same position in the second matrix.

To check for equality, we compare each element in both matrices position by position. If all corresponding elements match, the matrices are equal; if even one pair does not match, they are not equal. Understanding this concept is crucial in many applications of linear algebra, where accurate matrix operations are essential.

2x2 matrices

A 2x2 matrix is one of the simplest forms of matrices, with only two rows and two columns. Despite their simplicity, 2x2 matrices are quite powerful and can be used in a variety of mathematical and real-world problems. Each element in a 2x2 matrix can be referred to by its position. Common positions are

• (1,1): the top-left corner
• (1,2): the top-right corner
• (2,1): the bottom-left corner
• (2,2): the bottom-right corner

These matrices are easy to visualize and manage. Since there are only four elements, operations like addition, subtraction, and comparison are relatively simple compared to larger matrices. That's why they're often used as introductory examples in many math problems.

Matrix dimensions

Matrix dimensions are a fundamental aspect of understanding and working with matrices. The dimension of a matrix denotes the number of rows and columns it contains and is usually described as 'm by n' (written as \( m \times n \)), where \( m \) is the number of rows and \( n \) is the number of columns.
For example, a 2x2 matrix has 2 rows and 2 columns. Knowing the dimensions allows us to perform certain operations or comparisons. Two matrices must have the same dimensions in order to be added, subtracted, or compared for equality.Here are some key aspects of matrix dimensions:

• Adding or Subtracting Matrices: These operations are only possible if the matrices involved have the same dimensions.
• Equality: As previously noted, two matrices can't be equal unless their dimensions are identical.
• Matrix Multiplication: In this operation, the number of columns in the first matrix must match the number of rows in the second.

Matrix dimensions help us determine compatibility for these operations, making it a crucial concept in matrix algebra.