Problem 31 - Algebra 2

Question

In which matrix is the value of \(a_{32}\) less than the value of \(a_{21} ?\) F. \(\left[\begin{array}{rrr}{-1} & {0} & {5} \\ {4} & {3} & {-1} \\ {-3} & {2} & {6}\end{array}\right]\) G. \(\left[\begin{array}{rrr}{-1} & {5} & {0} \\ {3} & {4} & {-1} \\ {-3} & {6} & {2}\end{array}\right]\) H. \(\left[\begin{array}{rrr}{0} & {5} & {-1} \\ {-1} & {4} & {3} \\ {6} & {2} & {-3}\end{array}\right]\) J. \(\left[\begin{array}{rrr}{0} & {-1} & {5} \\ {0} & {3} & {4} \\ {-3} & {1} & {6}\end{array}\right]\)

Step-by-Step Solution

Answer

The value of \(a_{32}\) is less than the value of \a_{21}\) in Matrix F.

Step 1: Identify the Required Entries

The task is to find the matrix where the entry at the third row and second column (\(a_{32}\)) is less than the entry at the second row and first column (\(a_{21}\)). We need to individually check the values of these positions within each matrix option.

Step 2: Compare the Entries in Matrix F

In Matrix F, the value of the entry at the third row and second column is 2 (\(a_{32} = 2\)) and the value of the entry at the second row and first column is 4 (\(a_{21} = 4\)). Since 2 is less than 4, Matrix F satisfies the condition that \(a_{32} < a_{21}\).

Step 3: Verify the Remaining Matrices

For completeness, we check the other matrices as well. In Matrix G, \(a_{32} = 6\) and \(a_{21} = 3t\text{In Matrix H, }a_{32} = 2\) and \(a_{21} = -1t\text{In Matrix J, }a_{32} = 1\) and \(a_{21} =0 \).None of these matrices have \(a_{32}\) less than \(a_{21}\), so they do not satisfy the condition.

Key Concepts

Algebraic Matrices

When dealing with algebraic matrices, it's crucial to understand that they are essentially a rectangular array of numbers, symbols, or expressions arranged in rows and columns. These matrices are a fundamental concept in linear algebra and have applications in various areas such as physics, computer science, and economics.

Each entry in the matrix is called an element, and the position of each element is given by two indices — the row number and the column number. The notation for a matrix typically uses uppercase letters (like A, B, C), and its elements with corresponding lowercase letters along with two subscripted indices (like a_{ij}, where i denotes the row, and j represents the column).

Matrices can be added, subtracted, and multiplied by specific rules, which make them incredibly useful for solving systems of linear equations and for transforming geometric objects.

Matrix Elements

The individual items within a matrix are known as matrix elements or entries, referenced by their location within the matrix. Each element is denoted by a letter (often the matrix's letter in lowercase) followed by two subscripts: the first indicating the row and the second indicating the column.

For instance, in a matrix 'A', the element in the third row and second column is represented as a_{32}. It's critical for students to locate these elements accurately because they form the basis of matrix operations and comparisons. For example, when solving a system of linear equations represented by matrices, the values of these elements are directly used to find the solution.

Understanding the position and notation of matrix elements is a key skill in algebra that allows students to navigate through more complex matrix operations and applications.

Inequalities in Matrices

Inequalities in matrices come into play when we wish to compare the individual elements of one matrix with another, or within the same matrix, as seen in the textbook exercise. However, it is essential to note that we do not compare matrices as a whole using inequalities but rather their corresponding elements.

The concept is similar to comparing numbers but is done element-wise. If we have two matrices, A and B, of the same dimensions, we say that A is less than B (A < B) if and only if every corresponding element of A is less than the corresponding element in B.

In the context of the exercise, we are comparing specific positions within different matrices. The comparison involves looking at the value of the element in the third row and second column (a_{32}) and checking if it is less than the value in the second row and first column (a_{21}). Such comparisons might be used in optimization problems or in testing conditions for certain kinds of matrix transformations.