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

For (a) \(m = 4\) and for (b) \(n = 3\). Matrix \(B\) is \(4 \times 3\).

Step 1: Determine dimensions required for AB

For the matrix product \(AB\) to be defined, the number of columns in matrix \(A\) must be equal to the number of rows in matrix \(B\). Matrix \(A\) is a \(3 \times 4\) matrix, so it has 4 columns. Therefore, \(B\) must have 4 rows, which means \(m = 4\).

Step 2: Determine dimensions required for BA

For the matrix product \(BA\) to be defined, the number of columns in matrix \(B\) must be equal to the number of rows in matrix \(A\). Matrix \(A\) is a \(3 \times 4\) matrix, so it has 3 rows. Therefore, \(B\) must have 3 columns, which means \(n = 3\).

Step 3: Conclusion on dimensions of B

Combining the results from Step 1 and Step 2, we find that for both products to be defined, matrix \(B\) must be a \(4 \times 3\) matrix.

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is fundamental in linear algebra, especially when dealing with operations like matrix multiplication. Matrix dimensions are denoted as rows by columns, indicating how many rows and columns a matrix has. For example, a matrix with 3 rows and 4 columns is written as a 3x4 matrix. The significance of matrix dimensions becomes clear when determining whether two matrices can be multiplied.

In matrix multiplication, the number of columns in the first matrix must match the number of rows in the second matrix. This requirement is essential because each element in the resulting matrix is the sum of products of corresponding elements from the rows of the first matrix and columns of the second matrix. Therefore, to multiply matrices correctly, we need to ensure that their dimensions are compatible.

Matrix Product

The matrix product, more commonly known as matrix multiplication, involves taking two matrices and generating a third matrix from them. This process is different from simply multiplying numbers, as matrix multiplication involves a more structured approach.

Matrix multiplication is only possible when the number of columns in the first matrix matches the number of rows in the second. Let's take an example: if matrix \(A\) is 3x4 and matrix \(B\) is 4x2, we can multiply \(A\) and \(B\). However, if \(B\) were instead 5x2, multiplication would not be possible due to misaligned dimensions.

When the conditions for matrix multiplication are met, the resulting matrix from the product \(AB\), in this example, would be a 3x2 matrix. The new matrix's first element is calculated by taking the first row of matrix \(A\) and the first column of matrix \(B\), multiplying the respective elements, and summing them up. This element-wise multiplication and summation continue across all corresponding row and column pairs until the resulting matrix is filled.

Linear Algebra

Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between these spaces. It is the foundation for many mathematical theories and applications, particularly in understanding matrix operations and their implications.

Matrices are a primary component of linear algebra, used to represent linear transformations and systems of linear equations. Multiplying matrices, a key operation in linear algebra, forms the basis for complex computations in fields like machine learning, physics, and economics. This concept is vital because it allows us to model and solve systems of equations where multiple variables interact simultaneously.

• Matrix operations such as addition, subtraction, and multiplication enable transformations and manipulations within vector spaces.
• Understanding how to correctly multiply matrices requires a solid grasp of their dimensions and how they interact during multiplication.

Linear algebra's power lies in its ability to generalize and solve problems across various dimensions, making it an integral tool not just in mathematics, but in numerous scientific and engineering disciplines as well.