Problem 1 - Algebra and Trigonometry

Question

Determine whether the matrices \(A\) and \(B\) are equal. $$ A=\left[\begin{array}{rrr}{1} & {-2} & {0} \\ {\frac{1}{2}} & {6} & {0}\end{array}\right], \quad B=\left[\begin{array}{rr}{1} & {-2} \\\ {\frac{1}{2}} & {6}\end{array}\right] $$

Step-by-Step Solution

Answer

Matrices \(A\) and \(B\) are not equal because they have different dimensions.

Step 1: Understand the Definition of Matrix Equality

Two matrices are equal if and only if they have the same dimensions and corresponding elements are equal.

Step 2: Compare Dimensions of Matrices

Matrix \(A\) is a \(2 \times 3\) matrix (2 rows and 3 columns), while matrix \(B\) is a \(2 \times 2\) matrix (2 rows and 2 columns).

Step 3: Conclude Based on Dimension Comparison

Since matrices \(A\) and \(B\) have different dimensions, they cannot be equal. For two matrices to be equal, they must have identical dimensions and corresponding elements must be equal.

Key Concepts

Understanding Matrix Dimensions

When comparing matrices, the first step is to look at their dimensions, which define the shape and size of the matrix. The dimensions of a matrix are given as \(m \times n\), where \(m\) is the number of rows, and \(n\) is the number of columns.
For example, if you have a matrix \(A\) with dimensions \(2 \times 3\), it means that matrix \(A\) has 2 rows and 3 columns. Similarly, a matrix \(B\) with dimensions \(2 \times 2\) means it has 2 rows and 2 columns.
When comparing matrices for equality, the first rule is that they must have the same dimensions. If the number of rows or columns does not match, the matrices cannot be equal, regardless of the values they contain. This is crucial because even if a few elements of the matrices seem to match, differing dimensions make them inherently different structures.

The Process of Element Comparison

Once matrix dimensions are established as equal, the next step is element comparison. Each element in one matrix should correspond exactly to an element in the other matrix, both in terms of position and value.
Consider a matrix \(C\) with dimensions \(3 \times 2\) and another matrix \(D\) also with dimensions \(3 \times 2\). For \(C\) and \(D\) to be considered equal, the element in the first row, first column of \(C\) must be the same as the element in the first row, first column of \(D\), and so on for every element in the matrices.

• If at least one element differs in value, the matrices are not equal.
• The position of the elements is as important as their values. A mismatch in positions means the matrices are not equal.

Without passing the dimension equality first, this step is redundant because differing dimensions already conclude inequality.

What Makes Matrices Equal?

To determine if two matrices are equal, both the dimensions and elements need to match perfectly.
This means:

• The matrices must have the same number of rows.
• The matrices must have the same number of columns.
• Each corresponding element must be identical in both matrices.

Using the example from the original exercise:
Matrix \(A\) is \(2 \times 3\), while matrix \(B\) is \(2 \times 2\). Right from assessing dimensions, it's clear they can't be equal as they don't even have the same number of columns.
Even if we looked further into their elements, which we don't need to here because of the differing dimensions, matching elements still wouldn't make up for the difference in structure. Equal matrices must fulfill both these criteria: identical size and identical corresponding elements. This ensures the matrices represent the exact same data in structure and content.