Problem 28 - Calculus for Biology and Medicine

Question

Suppose \(A\) is a \(3 \times 4\) matrix and \(B\) is an \(m \times n\) matrix. What are values of \(m\) and \(n\) such that the following products are defined? (a) \(A B\) (b) \(B A\)

Step-by-Step Solution

Answer

(a) Matrix B must be \(4 \times n\); (b) Matrix B must be \(m \times 3\).

Step 1: Understanding Matrix Multiplication

For matrices to be multiplied, the number of columns in the first matrix must be equal to the number of rows in the second matrix. This is key in determining the dimensions needed for matrix multiplication.

Step 2: Determining Dimensions for (a) AB

Matrix A has dimensions \(3 \times 4\). For \(AB\) to be defined, the matrix B must have 4 rows. Thus, the dimension of B must be \(4 \times n\), where \(n\) can be any positive integer. The resulting matrix will have dimensions \(3 \times n\).

Step 3: Determining Dimensions for (b) BA

Matrix A has dimensions \(3 \times 4\). For \(BA\) to be defined, the matrix B must have 3 columns. Thus, the dimension of B must be \(m \times 3\), where \(m\) can be any positive integer, and the resulting matrix will have dimensions \(m \times 4\).

Key Concepts

Matrix Dimensions

Matrix dimensions are crucial in determining whether two matrices can be multiplied together. Every matrix is defined by its dimensions, which are indicated as rows × columns. For instance, a matrix with dimensions 3 × 4 has 3 rows and 4 columns. Understanding these dimensions is important because they dictate the shape and structure of the matrix.

When it comes to matrix multiplication, the inner dimensions — the number of columns in the first matrix and the number of rows in the second matrix — must match for the operation to be valid. This means if you have a matrix A with dimensions 3 × 4, it can only be multiplied by another matrix B if B has 4 rows, making the multiplication possible. If these conditions are not met, the matrix product cannot be computed.

Matrix Product

The matrix product, also known as matrix multiplication, results in a new matrix formed by the dot product of the rows of the first matrix with the columns of the second matrix. To perform matrix multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix, ensuring that every element in the resulting matrix is defined.

For example, consider two matrices: matrix A with dimensions 3 × 4 and matrix B with dimensions 4 × n. They can be multiplied to form a matrix product AB, which will have dimensions 3 × n. This new matrix is constructed by calculating the sum of products of corresponding elements from the rows of A and columns of B.

• Each element in the resulting matrix is a sum of products of elements from a row in A and a column in B.
• Matrix multiplication is not commutative, meaning AB may not equal BA.

This principle is fundamental in linear algebra and is widely used in various fields, including computer graphics, data science, and more.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, linear transformations, and matrices. It provides the necessary framework and tools for understanding structures such as systems of linear equations and transformations beyond basic arithmetic.

Matrices are a core component of linear algebra, acting as representations of complex systems in a structured, numerical format. Through the manipulation of matrices, linear algebra provides a means to easily perform operations like solving systems of equations, transforming geometric objects, and even encoding complex networks.

• Understanding matrix operations like multiplication helps in solving linear equations efficiently.
• Linear transformations, a key topic in linear algebra, often rely on matrix multiplication to map vectors from one space to another.

By mastering linear algebra, students unlock the ability to solve real-world problems, model scenarios in physics and engineering, and conduct data analysis in an elegant and powerful way.