Problem 2 - Algebra and Trigonometry

Question

Determine whether the matrices \(A\) and \(B\) are equal. $$ A=\left[\begin{array}{cc}{\frac{1}{4}} & {\ln 1} \\ {2} & {3}\end{array}\right], \quad B=\left[\begin{array}{cc}{0.25} & {0} \\\ {\sqrt{4}} & {\frac{6}{2}}\end{array}\right] $$

Step-by-Step Solution

Answer

The matrices \(A\) and \(B\) are equal.

Step 1: Compare Dimensions

First, check whether the dimensions of matrices \(A\) and \(B\) are the same. Both matrices \(A\) and \(B\) are 2x2 matrices, so they have the same dimensions. This is a necessary condition for matrix equality.

Step 2: Compare Each Element

Next, compare each corresponding element of matrices \(A\) and \(B\).- Element \(A_{11}\) is \(\frac{1}{4}\) and \(B_{11}\) is \(0.25\). Convert the fraction to decimal: \(\frac{1}{4} = 0.25\). The first elements are equal.- Element \(A_{12}\) is \(\ln 1\) and \(B_{12}\) is \(0\). Since \(\ln 1 = 0\), the second elements are equal.- Element \(A_{21}\) is \(2\) and \(B_{21}\) is \(\sqrt{4}\). Since \(\sqrt{4} = 2\), the third elements are equal.- Element \(A_{22}\) is \(3\) and \(B_{22}\) is \(\frac{6}{2}\). Since \(\frac{6}{2} = 3\), the fourth elements are equal.

Step 3: Conclude Equality

Since all corresponding elements of matrices \(A\) and \(B\) are equal, matrices \(A\) and \(B\) are equal matrices.

Key Concepts

Understanding Matrix Dimensions

When dealing with matrices, one of the first things to check is their dimensions. This tells us about the number of rows and columns in a matrix. For two matrices to be comparable or equal, they must have the same dimensions. If this criterion is not met, the matrices cannot be the same. For example, a 2x2 matrix has 2 rows and 2 columns. If we are told that matrix A and matrix B both are 2x2 matrices, we can proceed to the next step of checking their equality. Keep in mind these simple points:

• The dimensions are always presented as rows by columns, for instance, 2x2, 3x1, etc.
• Matrices that do not share the same dimensions cannot be compared for equality.

Element-wise Comparison of Matrices

Matrix equality involves comparing each corresponding element of the matrices in question. This means you look at the position of each element in matrix A and ensure it matches the element in the same position in matrix B.To compare elements:\[A_{ij}\] refers to the element in the i-th row and j-th column of matrix A. The goal is to check if each \(A_{ij}\) is equal to the corresponding \(B_{ij}\).Here are the steps:

• Go through each element in rows of matrix A and validate it against the same position in matrix B.
• Convert any fractions, logarithms, or square roots to their simplest form to facilitate accurate comparison.

Once each element in matrix A finds its exact counterpart in matrix B, we can say the matrices are equal. It’s crucial to examine every element methodically to ensure no discrepancies remain.

Exploring Matrices

Matrices are a fundamental concept in mathematics that find applications across various fields like physics, economics, and engineering. A matrix is essentially a set of numbers arranged in rows and columns, creating a rectangular array.Here are a few core concepts related to matrices:

• Matrices can represent data or perform transformations in any given context.
• Each entry in a matrix is known as an element, often noted as \(A_{ij}\) indicating its position within the matrix.
• Matrices come in various sizes and can be added, subtracted, and multiplied under certain conditions.

Understanding matrices is all about comprehending how these elements fit together to form a larger mathematical picture. By learning the basics of matrices, such as dimensions, operations, and comparisons, you unlock the ability to address complex mathematical problems seamlessly.