Problem 5 - 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 due to different dimensions.

Step 1: Compare Dimensions

First, we need to compare the dimensions of matrices \( A \) and \( B \). Matrix \( A \) has dimensions \( 2 \times 3 \) because it has 2 rows and 3 columns. Matrix \( B \) has dimensions \( 2 \times 2 \) because it has 2 rows and 2 columns. Matrices can only be equal if they have the same dimensions.

Step 2: Check if Dimensions Match

Since matrix \( A \) is \( 2 \times 3 \) and matrix \( B \) is \( 2 \times 2 \), they do not have the same dimensions. Therefore, \( A \) and \( B \) cannot be equal.

Key Concepts

Matrix Dimensions

When discussing matrices, one of the first things to consider is their dimensions. The dimensions of a matrix are defined by the number of its rows and columns. When we refer to a matrix with dimensions "2 x 3," it means the matrix has 2 rows and 3 columns.
This measurement is crucial because it helps identify the size and shape of the matrix, which is the first step in many matrix operations. To compare matrices, as shown in the problem, checking dimensions is often the first step to determining equality or suitability for operations like addition and multiplication.

• If matrices do not have the same dimensions, they cannot be equal or directly added or subtracted.
• The dimensions need to match exactly for two matrices to be equal.

In our example, matrix \(A\) has dimensions 2 x 3, while matrix \(B\) holds dimensions of 2 x 2. Hence, they differ in dimensions, making it impossible for them to be equal.

Matrix Comparison

Matrix comparison involves testing whether two matrices are identical. For two matrices to be considered equal:

• They must have the same dimensions, meaning an equal number of rows and columns.
• Each corresponding element needs to be the same.

In our given example, we quickly identify the mismatch in dimensions when comparing matrices \(A\) and \(B\). With matrix \(A\) being 2x3 and matrix \(B\) being 2x2, they clearly do not meet the first criteria of having equal dimensions. Therefore, there is no need to proceed to element comparison as they can never be equal.
This step saves time since identifying dimension mismatches quickly disqualifies the possibility of matrix equality.

Matrix Properties

Understanding the properties of matrices is essential for working with them effectively. Several key properties define and characterize matrices, influencing operations and comparisons.

**Key Matrix Properties include:**

• **Dimension:** as previously discussed, it's the most basic property defining the rows and columns.
• **Element uniqueness:** Each position in a matrix holds a specific value. For equality, every corresponding position in two matrices must contain identical elements.
• Commutative Property of Addition: For addition, matrices of the same dimension can add up in any sequence.
• Associative and Distributive Properties: These apply for both multiplication and addition within matrix operations.

Properties like these set the foundation for how matrices function and interact with one another. In the context of equality, ensuring each of these properties aligns between two matrices can determine their equivalence or lack thereof.