Problem 2 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The sizes of matrices \(A\) and \(B\) are given. Find the size of \(A B\) and \(B A\) whenever they are defined. \(A\) is of size \(3 \times 4\), and \(B\) is of size \(4 \times 3\).

Step-by-Step Solution

Answer

The size of \(AB\) is \(3 \times 3\) and the size of \(BA\) is \(4 \times 4\).

Step 1: Matrix Multiplication Requirements

To multiply two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. If this condition is met, then the product matrix exists, and its size will be the number of rows of the first matrix times the number of columns of the second matrix.

Step 2: Check if \(AB\) is defined

To check if \(AB\) is defined, we need to compare the number of columns in \(A\) with the number of rows in \(B\). The matrix \(A\) is of size \(3 \times 4\), so it has 3 rows and 4 columns. The matrix \(B\) is of size \(4 \times 3\), so it has 4 rows and 3 columns. Since the number of columns in \(A\) (4) matches the number of rows in \(B\) (4), \(AB\) is defined.

Step 3: Find the size of \(AB\)

Since \(AB\) is defined, we can now find its size. The size of the product matrix will be the number of rows in \(A\) times the number of columns in \(B\). \(A\) has 3 rows and \(B\) has 3 columns, so the size of \(AB\) will be \(3 \times 3\).

Step 4: Check if \(BA\) is defined

To check if \(BA\) is defined, we need to compare the number of columns in \(B\) with the number of rows in \(A\). The matrix \(B\) is of size \(4 \times 3\), so it has 3 columns. The matrix \(A\) is of size \(3 \times 4\), so it has 3 rows. Since the number of columns in \(B\) (3) matches the number of rows in \(A\) (3), \(BA\) is defined.

Step 5: Find the size of \(BA\)

Since \(BA\) is defined, we can now find its size. The size of the product matrix will be the number of rows in \(B\) times the number of columns in \(A\). \(B\) has 4 rows and \(A\) has 4 columns, so the size of \(BA\) will be \(4 \times 4\).

Step 6: Summary

The sizes of matrices \(AB\) and \(BA\) are given as follows: - \(AB\) is of size \(3 \times 3\) - \(BA\) is of size \(4 \times 4\)

Key Concepts

Matrix Dimensions

When we talk about matrix dimensions, we are referring to the number of rows and columns a matrix contains. This is often expressed in the format \( m \times n \), where \( m \) is the number of rows and \( n \) is the number of columns. For instance, a matrix of size \( 3 \times 4 \) has 3 rows and 4 columns. It's important to understand this because the dimensions help us determine other properties of matrices, such as whether they can be multiplied together.

• If you know the dimensions, you can tell if two matrices can be multiplied.
• The dimensions also tell you the size and shape of the resulting product when multiplication is possible.

Identifying matrix dimensions is the first step in working with matrices, particularly in operations like matrix multiplication.

Matrix Size

The term matrix size often goes hand in hand with matrix dimensions, but it emphasizes the spatial allocation of a matrix. A useful visualization is to think of the matrix as a grid, where the size determines the grid's arrangement. Matrix size affects operations that can be performed:

• The number of possible operations, such as addition, subtraction, and multiplication, depend on the matrices' sizes.
• Resizing a matrix is not a trivial operation, as it may require filling or trimming values to fit the new dimensions.

When considering the matrix size, it's crucial for computational efficiency and determining the types of mathematical operations applicable to the matrix data.

Matrix Product

The matrix product refers to the result of multiplying two matrices. Not every pair of matrices can be multiplied; specific conditions must be met. To multiply matrix \( A \) and matrix \( B \), the number of columns of \( A \) must equal the number of rows of \( B \).

• If these conditions are met, the resulting matrix's size is determined by the number of rows of \( A \) and the number of columns of \( B \).
• The entries of the resulting matrix are calculated by taking the dot product of the rows of \( A \) and columns of \( B \).

Understanding how to properly compute the matrix product is essential in many areas of mathematics and related fields.

Matrices

Matrices are fundamental in linear algebra, used to arrange numbers or symbols in rectangular grids. Each number in this grid is called an element. Matrices can represent a wide range of data and operations in mathematics, science, and engineering.

• Matrices can be added or subtracted when they have the same dimensions.
• They can also be multiplied with each other, given the correct conditions are fulfilled.
• They are instrumental in solving systems of linear equations and in performing transformations in graphics.

Understanding matrices is crucial for tackling more complex mathematical concepts, serving as a building block for many real-world applications.

Linear Algebra

Linear algebra is the branch of mathematics concerning linear forms such as lines, planes, and subspaces. It involves studying matrices and systems of linear equations. Linear algebra forms the basis for various fields like physics, computer science, and economics.

• It helps in solving simultaneous linear equations using matrix operations.
• The concepts of vectors and matrices are fundamental in understanding spaces and transformations.
• Linear algebra is essential in algorithms, machine learning, and data analysis.

Having a grasp on linear algebra concepts opens doors to advanced technological solutions and a deeper understanding of their mathematical foundations.