Problem 10 - Advanced Engineering Mathematics

Question

In Problems 7-10, determine whether the given matrices are equal. $$ \left(\begin{array}{rr} \frac{1}{8} & \frac{1}{5} \\ \sqrt{2} & 1 \end{array}\right),\left(\begin{array}{rr} 0.125 & 0.2 \\ 1.414 & 1 \end{array}\right) $$

Step-by-Step Solution

Answer

The matrices are not equal; \(\sqrt{2}\) is not exactly 1.414.

Step 1: Understand Matrix Equality

Matrix equality means that two matrices are identical, meaning they have the same dimensions and each corresponding element in the matrices is equal.

Step 2: Confirm Matrix Dimensions

Verify that both matrices have the same dimensions. Each matrix given is a 2x2 matrix. Since their dimensions match, we can proceed to check the individual elements.

Step 3: Compare Corresponding Elements

Compare each element in the first matrix with the corresponding element in the second matrix:1. Compare \(\frac{1}{8}\) with 0.125. \(\frac{1}{8} = 0.125\), so they are equal.2. Compare \(\frac{1}{5}\) with 0.2. \(\frac{1}{5} = 0.2\), so they are equal.3. Compare \(\sqrt{2}\) with 1.414. Since \(\sqrt{2}\) is approximately 1.414213, this is not exactly 1.414, hence these elements are not equal.4. Compare 1 with 1. They are equal.

Step 4: Determine if Matrices are Equal

For the matrices to be equal, all corresponding elements must match exactly. Since \(\sqrt{2}\) is not precisely 1.414, the matrices are not equal.

Key Concepts

Matrix Dimensions

When dealing with matrices, the first step to understanding their equality is to check their dimensions. Matrices are essentially rectangular arrays of numbers, organized in rows and columns. The dimensions of a matrix are given as "rows x columns". For instance, a matrix with 2 rows and 3 columns is a 2x3 matrix.
When comparing two matrices, it is crucial that they possess identical dimensions.

• If one matrix is a 2x2 and the other is a 3x3, they are immediately deemed unequal.
• Both matrices in this exercise are 2x2, meaning each has 2 rows and 2 columns, allowing us to proceed to the next step.

Consistent dimensions ensure that each element in one matrix has a corresponding element in the other matrix. So, always start by confirming the dimensions before moving on to other checks.

Comparing Matrices

Once you've confirmed that two matrices share the same dimensions, it's time to dive deeper, configuring your gaze to the details within. Direct comparison involves checking each corresponding element in the matrices.
For two matrices to be equal, every individual component must match its counterpart identically. In basic terms, you're lining up elements and doing a side-by-side comparison:

• Start from the top-left corner, comparing each element in the corresponding position.
• Move systematically across each row and down each column.

As each element is examined, you're looking for exact equality in value. Encountering even a single mismatch means the matrices aren’t equal, stressing the necessity of thoroughness in this process.

Matrix Elements

The element is the fundamental unit in a matrix, a small number that contributes to the whole. When comparing matrix elements, it’s vital to understand that these aren’t just numbers; they can take many forms, such as fractions or decimals.
In this exercise:

• We compared \(\frac{1}{8}\) with 0.125. Since both are equal (as \(\frac{1}{8}\) equals 0.125), we confirmed these elements match.
• Similarly, \(\frac{1}{5}\) was checked against 0.2 and found to be equal.
• Issues arose with the comparison of \(\sqrt{2}\) and 1.414. While 1.414 is an approximation, \(\sqrt{2}\) is exactly 1.414213562..., causing a mismatch here.

Such discrepancies, especially involving rounded values like square roots or repeating decimals, highlight the need for precision.
Therefore, while comparing, ensure absolute precision as even minor differences can render two matrices unequal.