Problem 3 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Refer to the following matrices: \(A=\left[\begin{array}{rrrr}2 & -3 & 9 & -4 \\ -11 & 2 & 6 & 7 \\ 6 & 0 & 2 & 9 \\ 5 & 1 & 5 & -8\end{array}\right]\) \(B=\left[\begin{array}{rrr}3 & -1 & 2 \\ 0 & 1 & 4 \\ 3 & 2 & 1 \\ -1 & 0 & 8\end{array}\right]\) \(C=\left[\begin{array}{lllll}1 & 0 & 3 & 4 & 5\end{array}\right]\) \(D=\left[\begin{array}{r}1 \\ 3 \\ -2 \\ 0\end{array}\right]\) Find \(b_{13}, b_{31}\), and \(b_{43}\).

Step-by-Step Solution

Answer

In matrix B, the values of \(b_{13}, b_{31}\), and \(b_{43}\) are as follows: \(b_{13} = 2\), \(b_{31} = 3\), and \(b_{43} = 8\).

Step 1: Locate \(b_{13}\) in Matrix B

In matrix B, we want to find the value in the first row (1) and the third column (3). Here's the matrix B: $B=\left[\begin{array}{rrr} 3 & -1 & 2 \\ 0 & 1 & 4 \\ 3 & 2 & 1 \\ -1 & 0 & 8 \end{array}\right]$ The value in the first row and the third column is 2, so \(b_{13} = 2\).

Step 2: Locate \(b_{31}\) in Matrix B

In matrix B, we want to find the value in the third row (3) and the first column (1). Here's the matrix B: $B=\left[\begin{array}{rrr} 3 & -1 & 2 \\ 0 & 1 & 4 \\ 3 & 2 & 1 \\ -1 & 0 & 8 \end{array}\right]$ The value in the third row and the first column is 3. Thus, \(b_{31} = 3\).

Step 3: Locate \(b_{43}\) in Matrix B

In matrix B, we want to find the value in the fourth row (4) and the third column (3). Here's the matrix B: $B=\left[\begin{array}{rrr} 3 & -1 & 2 \\ 0 & 1 & 4 \\ 3 & 2 & 1 \\ -1 & 0 & 8 \end{array}\right]$ The value in the fourth row and the third column is 8. Therefore, \(b_{43} = 8\).

Step 4: Final Results

Based on our step-by-step analysis of matrix B, we can conclude the following: - \(b_{13} = 2\) - \(b_{31} = 3\) - \(b_{43} = 8\)

Key Concepts

Matrix Elements

Matrix elements are the individual items or numbers within a matrix. Each element in a matrix is identified by its position, denoted by a double subscript system. For example, in a matrix B, a specific element is indicated as \(b_{ij}\), where \(i\) is the row number, and \(j\) is the column number.
Let's break it down further:

• The first subscript \(i\) refers to the row number.
• The second subscript \(j\) refers to the column number.

In the exercise provided, the elements \(b_{13}\), \(b_{31}\), and \(b_{43}\) from matrix B were located using their respective row and column numbers.
Understanding how to identify and interpret matrix elements is crucial to progressing in more complex matrix operations.

Matrix Notation

Matrix notation is a structured way of presenting numbers in rows and columns neatly arranged in a rectangular array. This notation is a key concept in linear algebra, allowing for concise representation and communication of complex operations.
For instance, consider matrix B from our exercise:

• Matrix B is a 4x3 matrix, which means it has 4 rows and 3 columns.
• Each number within this matrix is an element, as discussed earlier.

Matrices are usually represented with capital letters like A, B, C, etc. The elements within a matrix are accessed using their specific notation, such as \(b_{ij}\) indicating the element at the \(i\)-th row and \(j\)-th column.
Using matrix notation not only helps in identifying matrix dimensions quickly but also in performing matrix operations succinctly.

Row and Column Identification

Rows and columns are the building blocks of a matrix, defining its structure and dimensions. Identifying and understanding rows and columns is essential for any operation involving matrices.

Rows

• Rows are the horizontal lines of elements in a matrix.
• For example, in matrix B, the first row is \([3, -1, 2]\).

Columns

• Columns are the vertical lines of elements.
• In the same matrix, the first column is \([3, 0, 3, -1]\).

Being able to identify a specific row or column is crucial when locating particular elements, performing row or column operations, or even when determining the size of the matrix.Remember that in the notation \(b_{ij}\), \(i\) denotes the row, and \(j\) denotes the column the element belongs to. This makes row and column identification critical for accessing and correctly interpreting the elements in any matrix.