Problem 8 - Advanced Engineering Mathematics

Question

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.

Step 1: Understand Matrix Equality

Two matrices are equal if and only if they have the same dimensions and each corresponding element in the matrices is equal.

Step 2: Compare Dimensions of Matrices

Both matrices \( A = \left(\begin{array}{ll} 1 & 2 \ 0 & 1 \end{array}\right) \) and \( B = \left(\begin{array}{ll} 1 & 0 \ 2 & 1 \end{array}\right) \) are 2x2 matrices, so they have the same dimensions.

Step 3: Compare Corresponding Elements

Compare each element at the same position:- Top-left: \( A_{11} = 1 \) and \( B_{11} = 1 \). They are equal.- Top-right: \( A_{12} = 2 \) and \( B_{12} = 0 \). They are not equal.- Bottom-left: \( A_{21} = 0 \) and \( B_{21} = 2 \). They are not equal.- Bottom-right: \( A_{22} = 1 \) and \( B_{22} = 1 \). They are equal.

Step 4: Determine Result of Comparison

Since not all corresponding elements of the matrices are equal, the matrices are not equal.

Key Concepts

Matrix Comparison

Matrix comparison is a fundamental concept in linear algebra that helps us understand the relationships between two or more matrices. When comparing matrices, the primary goal is to check if the matrices are equal. This means we need to evaluate whether the matrices have the same dimensions and identical corresponding elements.

• Same dimensions: The matrices must have the same number of rows and columns to be considered for equality.
• Identical elements: Each corresponding element in the matrices should be equal.

Understanding this helps us analyze whether operations like matrix subtraction will result in a zero matrix, which is an important check in various mathematical applications. Ensuring that matrices meet these criteria before declaring them equal helps prevent errors in computations involving matrices.

Matrix Dimensions

Matrix dimensions refer to the size of the matrix, specifically the number of rows and columns it contains. For a matrix to be characterized as having dimensions "m x n," it must have 'm' rows and 'n' columns.
When comparing two matrices for equality:

• Both matrices must have the identical number of rows and columns.
• It is represented in the format m x n. For example, a 2x2 matrix has 2 rows and 2 columns.

The importance of having the same dimensions lies in setting up a consistent and valid comparison of each matrix element. If the dimensions differ, the matrices cannot be compared directly for equality. So, this step is often the first when determining matrix equality.

Element-wise Comparison

Element-wise comparison is the process of individually comparing each element in one matrix with the corresponding element in another matrix. This step follows after confirming that two matrices have identical dimensions.
For a thorough comparison, each element at the same position in the matrices must be evaluated. In mathematical terms:

• If Matrix A = \([a_{ij}]\) and Matrix B = \([b_{ij}]\), then for matrices to be equal, \(a_{ij} = b_{ij}\) for all i and j.
• Start from the top-left element and proceed in a systematic manner through rows and columns.

This meticulous approach helps in identifying exact matches and discrepancies between matrices. In our original exercise, comparing elements revealed differences in the second (top-right) and third (bottom-left) elements of the given matrices, leading to the conclusion that the matrices are not equal. This step of element-wise checking is crucial in ensuring accurate and precise matrix comparison.