Problem 5 - Precalculus Mathematics for Calculus

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: Check the Dimensions of Matrix A

Matrix \( A \) is a 2x3 matrix because it has 2 rows and 3 columns. Verify the dimensions as part of comparing two matrices.

Step 2: Check the Dimensions of Matrix B

Matrix \( B \) is a 2x2 matrix because it has 2 rows and 2 columns. It is important to note dimensions first since two matrices with different dimensions cannot be equal.

Step 3: Compare Dimensions of A and B

Compare the dimensions obtained in Steps 1 and 2. Since matrix \( A \) is 2x3 and matrix \( B \) is 2x2, they do not have the same dimensions.

Step 4: Conclude Equality Based on Dimensions

Matrices must have identical dimensions to be equal. Since \( A \) and \( B \) have different dimensions, they cannot be equal.

Key Concepts

Understanding Matrix Dimensions

A matrix is essentially a rectangular array of numbers arranged in rows and columns. The dimension of a matrix is crucial information that tells us how many rows and columns make up the matrix. This is usually expressed in the form \( m \times n \), where \( m \) stands for the number of rows and \( n \) denotes the number of columns. For example:

• If a matrix has 3 rows and 2 columns, its dimensions are \( 3 \times 2 \).
• A square matrix, like the 3x3 identity matrix, has equal numbers of rows and columns.

Knowing the dimensions helps in performing operations like addition, subtraction, and determining equality between two matrices. It's the first check you must carry out while comparing two matrices.

Matrix Comparison

When we're asked to determine if two matrices are equal, we need to perform a few checks:

• Check Dimensions: Two matrices can only be compared for equality if they have the same dimensions. If the number of rows and columns in the matrices differ, they cannot be equal.
• Element by Element Comparison: If the matrices share the same dimensions, we then compare each corresponding element. All elements must match for the matrices to be considered equal.

So, for matrices \( A \) and \( B \) in our problem, since they differ in dimensions (\( A \) is \( 2 \times 3 \) while \( B \) is \( 2 \times 2 \)), further comparison is unnecessary. The difference in dimensions alone means they cannot be equal.

Overview of Matrices

Matrices are an essential part of linear algebra and are used to represent and solve various mathematical problems. They can represent systems of equations, perform linear transformations, and are even used in computer graphics and machine learning. Here are a few key points:

• Structure: A matrix consists of rows and columns, and each element in the matrix is referred to by its position within these rows and columns.
• Types of Matrices: There are several types, including square matrices, rectangular matrices, identity matrices, and zero matrices, each having specific properties and uses.
• Applications: Beyond math, matrices are pivotal in science and engineering, serving as a framework for virtually everything from computer programming to physics simulations.

Understanding matrices, including their dimensions and how to compare them, is foundational knowledge necessary to leverage their full potential in both theoretical and practical applications.