Problem 10 - Advanced Engineering Mathematics

Question

Determine whether the given matrices are equal. $$ \left(\begin{array}{rr} \frac{1}{8} & \frac{1}{5} \\ \sqrt{2} & 1 \end{array}\right),\left(\begin{array}{lr} 0.125 & 0.2 \\ 1.414 & 1 \end{array}\right) $$

Step-by-Step Solution

Answer

The matrices are not equal because one corresponding pair of elements differ.

Step 1: Compare Dimensions

First, ensure that both matrices have the same dimensions. Both matrices are 2x2, meaning each has 2 rows and 2 columns. Thus, they can potentially be equal.

Step 2: Compare Corresponding Elements

Now, compare each corresponding element in the matrices to see if they are identical. - First row, first column: \(\frac{1}{8} = 0.125\) is true. - First row, second column: \(\frac{1}{5} = 0.2\) is also true.- Second row, first column: \(\sqrt{2} ≈ 1.414\). However, \(\sqrt{2}\) is actually approximately 1.41421356, so these elements are not equal. - Second row, second column: \(1 = 1\), which is true.

Step 3: Conclusion on Matrix Equality

Since not all corresponding elements from both matrices are equal, the matrices are not equal. Specifically, the elements in the second row, first column differ slightly.

Key Concepts

Understanding Matrix Comparison

When comparing matrices, the first step is to determine if two matrices are equal. Matrix comparison involves checking if both matrices are not only the same size but also if each corresponding element within them is identical. In mathematical terms, two matrices are equal if:

• They have the same number of rows and columns.
• Each element of one matrix is exactly the same as the corresponding element in the other matrix.

To affirm equality, examine each element in order, row by row and column by column, ensuring precision in each comparison. For example, fractions and their decimal equivalents must align precisely, as even the slightest deviation—as seen with approximations such as \( \sqrt{2} \)—can result in the matrices being deemed unequal.
Matrix comparison is integral in many applications of linear algebra, including solutions of linear systems and transformations.

Exploring Matrix Dimensions

Every matrix is fundamentally defined by its dimensions, which are given as the number of rows by the number of columns, expressed as 'm x n'. The dimension must be one of the primary checks when comparing matrices for equality, as only matrices of identical dimensions can potentially be equal. Here are essential points regarding matrix dimensions:

• A 2x2 matrix has 2 rows and 2 columns.
• Each element of a matrix occupies a distinct position identified by its row and column number (e.g., element at first row, first column).
• For two matrices to be compatible in operations like addition, subtraction, or equality checks, their dimensions must match.

Understanding matrix dimensions is crucial in matrix operations. This concept ensures clarity in solving problems involving matrices in mathematics and related fields.

Analyzing Matrix Elements

Matrix elements are the individual values that reside within a given matrix. Each element is located at a specific position described by the matrix's row and column structure. To determine if matrices are equal, you must compare their corresponding elements. Here’s how:

• Begin by comparing the first element of one matrix with the first element of the other matrix.
• Proceed in a systematic manner, element by element, until all elements have been compared.
• It is essential to pay attention to precision, especially when dealing with irrational numbers or approximations.

Even minor discrepancies between corresponding elements can result in inequality. For instance, while \( \sqrt{2} \) is often approximated as 1.414, the exact value is a non-repeating, non-terminating decimal. A small discrepancy like this can break the condition of equality. Accurate comparison is key in applications ranging from computer graphics to engineering calculations.

Introduction to Matrix Algebra

Matrix algebra forms the foundation for various mathematical operations that involve matrices, including addition, subtraction, multiplication, and equality checks. Understanding these operations is crucial to manipulating and applying matrices effectively. Key points about matrix algebra include:

• Matrix equality is just one aspect and requires both matching dimensions and identical elements.
• Matrix addition/subtraction requires matrices to have the same dimensions, with respective elements added or subtracted from one another.
• Matrix multiplication involves more complexity, where the number of columns in the first matrix must equal the number of rows in the second matrix.

Mastering matrix algebra unlocks numerous practical applications. From solving linear equations in physics to optimizations in computer science, matrix operations are fundamental tools in diverse scientific disciplines.