Problem 9 - Advanced Engineering Mathematics

Question

In Problems 7-10, determine whether the given matrices are equal. $$ \left(\begin{array}{cc} \sqrt{(-2)^{2}} & 1 \\ 2 & \frac{2}{8} \end{array}\right),\left(\begin{array}{rr} -2 & 1 \\ 2 & \frac{1}{4} \end{array}\right) $$

Step-by-Step Solution

Answer

Matrices are not equal.

Step 1: Understanding Matrix Equality

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

Step 2: Compare Dimensions

The given matrices are both 2x2 matrices. Since they have the same dimensions, we can move forward to compare individual elements.

Step 3: Compare Individual Elements

Start with the element in the first row, first column: \(\sqrt{(-2)^2}\) vs. \(-2\). Calculate \(\sqrt{(-2)^2} = \sqrt{4} = 2\). So, 2 is not equal to -2.

Key Concepts

Matrix Dimensions

Before determining whether two matrices are equal, it’s crucial to understand the concept of matrix dimensions. Matrices are essentially organized grids of numbers, and their dimensions tell us how many rows and columns they contain. In a matrix written as an 'm x n' type, 'm' signifies the number of rows, while 'n' indicates the number of columns.
For two matrices to be equal, they must share identical dimensions. This means, for example, that a 2x2 matrix can only be compared to another 2x2 matrix; a 2x3 matrix or a 3x2 matrix would be immediately disqualified from equality comparison with a 2x2 matrix.
In the problem given, both matrices are 2x2, meaning they each have 2 rows and 2 columns. Hence, they fulfill the initial requirement for matrix equality, allowing us to proceed to the next step.

Matrix Entries Comparison

Now that we know the two matrices share the same dimensions, the next step is to compare their corresponding entries. Each entry in a matrix has a position denoted by its row and column number. A key part of checking for matrix equality involves ensuring that every corresponding entry in the matrices is identical.
For instance, consider the position of the entry in the first row and first column in both matrices. In the given example, the matrices contain \( \sqrt{(-2)^2} \) in the first matrix and \( -2 \) in the second matrix. We calculate \( \sqrt{(-2)^2} \) which results in 2. Since 2 is not equal to -2, this proves that this pair of matrices is not equal.
It’s important to systematically verify each corresponding entry across all rows and columns, but as soon as one mismatch is found, the matrices are not equal.

Matrix Operations

Matrix operations refer to specific mathematical manipulations involving matrices. While our focus here is equality, understanding basic operations provides context for how matrices interact and combine. Common operations include addition, subtraction, and multiplication.
Addition and subtraction involve corresponding entries. For instance, given two matrices of the same dimension, you add or subtract the entries occupying the same position. Matrices of different dimensions cannot be directly added or subtracted.
Multiplication is slightly more complex. A matrix can only be multiplied by another if the number of columns in the first is equal to the number of rows in the second. This operation results in a new matrix. The element in the resulting matrix is a sum of products of corresponding entries from its respective row and column from the original matrices.
Understanding these operations is essential because they are foundational to more advanced concepts in linear algebra, including determining invertibility and performing transformations.