Problem 9 - Advanced Engineering Mathematics

Question

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

The matrices are not equal because their elements differ.

Step 1: Compare Dimensions

First, we need to check whether both matrices have the same dimensions. The given matrices are both 2x2 matrices. Since they have the same dimensions, we can proceed to compare their individual elements.

Step 2: Compare Corresponding Elements

We will now compare each corresponding element of both matrices.- Start with the top-left element: Compare \( \sqrt{(-2)^2} \) from the first matrix with \(-2\) from the second matrix.- The top-right elements are both \(1\).- Compare the bottom-left elements: Both are \(2\).- Finally, compare the bottom-right elements: \( \frac{2}{8} \) from the first matrix against \( \frac{1}{4} \) from the second matrix.

Step 3: Simplify and Validate Elements

Simplify and evaluate each element as needed to confirm equality.- For \( \sqrt{(-2)^2} \): The square of \(-2\) is \(4\), and \(\sqrt{4} = 2\). Compare this with \(-2\), but since \(2 eq -2\), the matrices differ at this element.- Check \( \frac{2}{8} \): Simplify to \( \frac{1}{4} \) (since \(2 \div 8 = 0.25\)), which matches the bottom-right element of the second matrix.

Step 4: Conclusion on Matrix Equality

Since the top-left elements of the matrices differ (\(2 eq -2\)), even after simplification, the matrices are not equal.

Key Concepts

Matrix Comparison

When we talk about matrices, one important concept is matrix comparison. This involves comparing two or more matrices to determine if they are equal.
Matrix equality means every corresponding element of the matrices must be the same. To check this, you need to look closely at each element in the matrices.

• Start by comparing the dimensions of the matrices, which is crucial to continue with any further comparisons.
• Once the dimensions match, focus on each corresponding element in both matrices.

In our example, even though the matrices had the same size (2x2), they differed in their elements. Particularly, the element \( \sqrt{(-2)^2} \) evaluated to 2, differed from the corresponding element \(-2\), making the matrices unequal. Being thorough with both dimensions and elements will lead you to the correct conclusion.

Matrix Elements

Understanding matrix elements is key to solving problems like the one given. Matrix elements are the individual numbers or expressions in a matrix. They are arranged in rows and columns, making up the entire matrix. In a 2x2 matrix, this means there are four elements in total.
To determine equality, each of these elements must match with the corresponding element in another matrix.

• In our example, we had to evaluate \( \sqrt{(-2)^2} \). This demonstrates the need sometimes to perform arithmetic simplifications on elements.
• We also considered \( \frac{2}{8} \) and simplified it to \( \frac{1}{4} \), matching with the element from the other matrix.

Every element needs individual attention; some require calculations before determining if the values are indeed the same.

Matrix Dimensions

Matrix dimensions are fundamental when comparing matrices. A matrix is defined by the number of its rows and columns. For matrix equality, both matrices must have the same dimensions.
For example, the dimensions of a matrix could be 2x2 - indicating two rows and two columns. If two matrices have different dimensions, they can't be compared directly on the basis of equality.

• In our problem, both matrices were of dimension 2x2, so we could proceed to compare elements.
• If the matrices had been 2x3 and 2x2, we would have concluded they were unequal without further checking.

Understanding dimensions ensures that you're on the right path in matrix comparison. It helps in identifying whether it's possible to proceed with checking each element or not.